What are the postulates that can be used to derive geometry? What are the various sets of postulates that can used to derive Euclidean geometry?
It might be nice to have several different approaches together for comparison purposes and for ready reference.
It might also be interesting to include an axiomatization (or two) of elliptical geometry.
 A: Axioms used in David Hilbert's The Foundations of Geometry (1899), as translated by E. J. Townsend in 1902
The undefined terms in Hilbert's axiomatization are point, line, plane, lies upon, between and congruent.
Group I: Axioms of Connection.
The axioms of this group establish a connection
between the concepts indicated above; namely, points,
straight lines, and planes. These axioms are as follows:


*

*Two distinct points $A$ and $B$ always completely
determine a straight line $a$. We write $AB = a$
or $BA = a$.

*Any two distinct points of a straight line completely
determine that line; that is, if $AB = a$ and
$AC=a$, where $B \neq C$, then is also $BC=a$.

*Three points $A$, $B$, $C$ not situated in the same
straight line always completely determine a plane
$\alpha$. We write $ABC=a$.

*Any three points $A$, $B$, $C$ of a plane $\alpha$, which
do not lie in the same straight line, completely determine
that plane.

*If two points $A$, $B$ of a straight line $a$ lie in
a plane $\alpha$, then every point of $a$ lies in $a$.

*If two planes $\alpha$, $\beta$ have a point $A$ in common,
then they have at least a second point $B$ in common.

*Upon every straight line there exist at least two
points, in every plane at least three points not
lying in the same straight line, and in space there
exist at least four points not lying in a plane.
Group II: Axioms of Order.
The axioms of this group define the idea expressed
by the word between, and make possible, upon the
basis of this idea, an order of sequence of the points
upon a straight line, in a plane, and in space. The
points of a straight line have a certain relation to one
another which the word between serves to describe.
The axioms of this group are as follows:


*

*If $A$, $B$, $C$ are points of a straight line and
$B$ lies between $A$ and $C$, then $B$ lies also between
$C$ and $A$.

*If $A$ and $C$ are two points of a straight line,
then there exists at least one point $B$ lying between
$A$ and $C$ and at least one point $D$ so situated that
$C$ lies between $A$ and $D$.

*Of any three points situated on a straight line,
there is always one and only one which lies between
the other two.

*Any four points $A$, $B$, $C$, $D$ of a straight line
can always be so arranged that $B$ shall lie between
$A$ and $C$ and also between $A$ and $D$, and, furthermore,
that $C$ shall lie between $A$ and $D$ and also
between $B$ and $D$

*Let $A$, $B$, $C$ be three points not lying in the
same straight line and
let $a$ be a straight line lying in the plane $ABC$ and not passing through any of the
points $A$, $B$, $C$. Then, if the straight line $a$ passes through a point
of the segment $AB$, it will also pass through either a point of the segment
$BC$ or a point of the segment $AC$.
Group III: Axiom of Parallels (The Axiom of Euclid).
The introduction of this axiom simplifies greatly
the fundamental principles of geometry and facilitates
in no small degree its development. This axiom may
be expressed as follows:


*

*In a plane $\alpha$ there can be drawn through any
point $A$, lying outside of a straight line $a$, one and
only one straight line which does not intersect the
line $a$. This straight line is called the parallel to
$a$ through the given point $A$.


Group IV. Axioms of Congruence.
The axioms of this group define the idea of congruence
or displacement.
Segments stand in a certain relation to one another
which is described by the word congruent.


*

*If $A$, $B$ are two points on a straight line $a$,
and if $A'$ is a point upon the same or another
straight line $a'$, then, upon a given side of $A'$ on
the straight line $a'$, we can always find one and
only one point $B'$ so that the segment $AB$ (or $BA$)
is congruent to the segment $A'B'$. We indicate
this relation by writing $AB\equiv A'B'$.
Every segment is congruent to itself; 
that is, we always have $AB\equiv AB$.

*If a segment $AB$ is congruent to the segment
$A'B'$ and also to the segment $A''B''$, then the segment
$A'B'$ is congruent to the segment $A''B''$; that
is, if $AB \equiv A'B$ and $AB \equiv A''B''$, then
$A'B' \equiv A''B''$.

*Let $AB$ and $BC$ be two segments of a straight
line $a$ which have no points in common aside from
the point $B$, and, furthermore, let $A'B'$ and $B'C'$
be two segments of the same or of another straight
line $a'$ having, likewise, no point other than $B'$ in
common.  Then, if $AB \equiv A'B'$ and $BC \equiv B'C'$,
we have $AC \equiv A'C'$.

*Let an angle $(h,k)$ be given in the plane
$\alpha$ and let a straight line $a'$ be given in a plane $\alpha'$.
Suppose also that, in the plane $\alpha$, a definite side
of the straight line $a'$ be assigned. Denote by $h'$ a
half-ray of the straight line $a'$ emanating from a
point $O'$ of this line. Then in the plane $\alpha'$ there
is one and only one half-ray $k'$ such that the angle
$(h,k)$, or $(k,h)$, is congruent to the angle $(h',k')$
and at the same time all interior points of the angle
$(h',k')$ lie upon the given side of $a'$. We express
this relation by means of the notation
$\angle (h,k) \equiv \angle (h',k')$ 
Every angle is congruent to itself; that is, 
$\angle (h,k) \equiv \angle (h,k)$
or $\angle (h,k) \equiv \angle (k,h)$.

*f the angle $(h,k)$ is congruent to the angle
$(h',k')$ and to the angle $(h'',k'')$, then the angle
$(h',k')$ is congruent to the angle $(h'',k'')$; that is
to say, if $\angle (h, k) \equiv \angle (h', k')$ and
$\angle (h, k) \equiv \angle (h'',k'')$, then
$\angle (h',k') \equiv \angle (h'',k'')$.

*If, in the two triangles $ABC$ and $A'B'C'$
the congruences
$AB \equiv A'B', \: AC \equiv A'C', \: \angle BAC \equiv \angle B'A'C'$
hold, then the congruences $\angle ABC \equiv \angle A'B'C' \:\mbox{and}\; \angle ACB \equiv \angle A'C'B'$
also hold.
Some definitions relevant to the axioms of congruence:
Let $\alpha$ be any arbitrary plane and $h$,
$k$ any two distinct half-rays lying in $\alpha$ and emanating
from the point $O$ so as to form a part of two different
straight lines. We call the system formed by these
two half-rays $h$, $k$ an angle and represent it by the
symbol $\angle(h, k)$ or $\angle(k, h)$. From axioms II, 1--5, it
follows readily that the half-rays $h$ and $k$, taken together
with the point $O$, divide the remaining points
of the plane a into two regions having the following
property: If $A$ is a point of one region and $B$ a point
of the other, then every broken line joining $A$ and $B$
either passes through $O$ or has a point in common
with one of the half-rays $h$, $k$. If, however, $A$, $A'$
both lie within the same region, then it is always possible
to join these two points by a broken line which
neither passes through $O$ nor has a point in common
with either of the half-rays $h$, $k$. One of these two
regions is distinguished from the other in that the segment
joining any two points of this region lies entirely
within the region. The region so characterised is
called the interior of the angle $(h,k)$. To distinguish
the other region from this, we call it the exterior of
the angle $(h,k)$. The half rays $h$ and $k$ are called the
sides of the angle, and the point $O$ is called the vertex
of the angle.
Group V. Axiom of Continuity (The Axiom of Archimedes).
This axiom makes possible the introduction into
geometry of the idea of continuity. In order to state
this axiom, we must first establish a convention concerning
the equality of two segments. For this purpose,
we can either base our idea of equality upon the
axioms relating to the congruence of segments and
define as equal the correspondingly congruent segments,
or upon the basis of groups I and II, we may
determine how, by suitable constructions, 
a segment is to be laid off from a point of a
given straight line so that a new, definite segment is
obtained equal to it. In conformity with such a
convention, the axiom of Archimedes may be stated
as follows:


*

*Let $A_1$ be any point upon a straight line between
the arbitrarily chosen points $A$ and $B$. Take the
points $A_2$, $A_3$, $A_4,\ldots$ so that $A_1$ lies between $A$
and $A_2$, $A_2$ between $A_1$ and $A_3$, $A_3$ between $A_2$ and
$A_4$ etc. Moreover, let the segments
$  A A_1, \; A_1 A_2, \; A_2 A_3, \; A_3 A_4, \;\ldots$
be equal to one another. Then, among this series
of points, there always exists a certain point $A_n$
such that $B$ lies between $A$ and $A_n$.

A: Postulates used in John M. Lee's Axiomatic Geometry (Draft, 2011) 
Used with author's permission
Postulates of Neutral Geometry
The undefined terms in Jack Lee's axiomatization are point, line, distance (between points) and measure (of an angle).
Postulate 1 (The Set Postulate). Every line is a set of points, and there is a set of point called the plane.
Postulate 2 (The Existence Postulate). There exist at least two distinct points.
Postulate 3 (The Unique Line Postulate). Given any two points, there is a unique line that contains both of them.
Postulate 4 (The Distance Postulate). For every pair of points $A$ and $B$, the distance between $A$ and $B$ is a non-negative real number determined by $A$ and $B$.
Postulate 5 (The Ruler Postulate). For every line $\ell$, there is a bijective function $\mathcal{f}:\ell \rightarrow \mathbb{R}$ with the property that for any two points $A, B \in \ell$, we have 
$$ AB = | \mathcal{f}(B) - \mathcal{f}(A)|.$$
Postulate 6 (The Plane Separation Postulate). For any line $\ell$, the set of all points not on $\ell$ is the union of two disjoint, non-empty subsets called sides of $\ell$. If $A$ anf $B$ are distinct points not on $\ell$, then both of the following conditions are satisfied:


*

*$A$ and $B$ are on the same side of line $\ell$ if and only if $\overline{AB} \cap \ell= \oslash$

*$A$ and $B$ are on the opposite side of line $\ell$ if and only if $\overline{AB} \cap \ell \neq \oslash$
Postulate 7 (The Angle Measure Postulate). For every angle $\angle ab$, the measure of $\angle ab$, written as $m\angle ab$, is a real number in the closed interval $[0,180]$ determined by $\angle ab$.


*

*Definition: If $\overrightarrow r$ is a ray starting at point $O$, and $P$ is a point not on the line $\overleftrightarrow r$, the half-rotation of rays determined by $\overrightarrow r$ and $P$, denoted by $HR(\overrightarrow r, P )$, is the set whose elements are all the rays that start at $O$ and whose points are either in line $\overleftrightarrow r$ or on the same side of $\overleftrightarrow r$ as $P$. 


Postulate 8 (The Protractor Postulate). For every ray $\overrightarrow r$ and every point $P$ not on line $\overleftrightarrow{r}$, there is a bijective function $g:HR(\overrightarrow{r}, P) \rightarrow [0,180]$ such that the following two conditions are satisfied:


*

*the function $g$ assigns the number $0$ to $\overrightarrow{r}$ and the number $180$ to the ray opposite $\overrightarrow{r}$. 

*If $\overrightarrow{a}$ and $\overrightarrow{b}$ are any two rays in $HR(\overrightarrow{r}, P)$ then $$ m\angle ab = |g(\overrightarrow b)-g(\overrightarrow b)|.$$
Postulate 9 (The SAS Postulate). If there is a correspondence between the vertices of two triangles such that two sides and the included angle of one triangle are congruent to the corresponding sides and angle of the other triangle, then the triangles are congruent under that correspondence.
Postulates of Euclidean Geometry
Postulates for Euclidean geometry are the postulates are Postulates 1–9 of neutral geometry, plus the following: 
Postulate 10E (The Euclidean Parallel Postulate). For each line $\ell$ and each point $A$ that does not lie on $\ell$, there is a unique line that contains $A$ and is parallel to $\ell$.


*

*Definition: Lines $\ell$ and $m$ are called parallel if they do not intersect.


There is one additional undefined term for Euclidean geometry: area.
Postulate 11E (The Euclidean Area Postulate).   For every polygonal region $\mathcal{R}$, there is a positive real number $\alpha(\mathcal{R})$ called the area of $\mathcal{R}$, which satisfies the following three conditions:


*

*(Area Congruence Property) If $\mathcal{R}_1$ and $\mathcal{R}_2$ are congruent simple regions, then $$ \alpha(\mathcal{R}_1)=\alpha(\mathcal{R}_2).$$

*(Area Addition Property) If $\mathcal{R}_1, \dots , \mathcal{R}_n$ are non-overlapping simple regions, then $$\alpha(\mathcal{R}_1 \cup \cdots \cup \mathcal{R}_n)= \alpha(\mathcal{R}_1) + \cdots + \alpha( \mathcal{R}_n).$$  

*(Unit Area Property) If $\mathcal{R}$ is a square region with sides of length $1$, then $\alpha(\mathcal{R})=1$.
Some useful definitions: 


*

*A polygon is a broken line segment that is simple closed and proper.

*A broken line segment is the union of finitely many segments $\overline{A_1 A_2},\, \overline{A_2 A_3}, \dots ,\, \overline{A_{n} A_{n+1}}$ determined by points $A_1, \dots , A_{n+1}$, not necessarily distinct. We denote this $\overline{A_1, \dots , A_{n+1}}$.

*Given a broken line segment $\overline{A_1, \dots , A_{n+1}}$, we call it 


*

*simple if the first $n$ vertices are distinct and no two of its constituent segments, edges, intersect except at a common end point.

*closed if the first and last points points, $A_1$ and $A_{n+1}$, are the same.

*proper if for each $i=1, \dots , n-1$, the three end points $A_i, A_{i+1}, A_{i+2}$ are non-collinear and if the broken line segment is closed, we require that $A_n, A_1,  A_2$ are non-collinear as well.


*We call two polygons congruent if there is a correspondence between their vertices such that consecutive vertices correspond to consecutive vertices, corresponding edges are congruent, and corresponding interior angle measures are equal.

*Given a polygon $\mathcal{P}$, a point $Q$ not on $\mathcal{P}$ is said to be an interior point of $\mathcal{P}$ if a ray starting with $Q$ and not containing any vertices of the polygon $\mathcal{P}$ has an odd number of intersections with the polygon.

*We call a set of points $\mathcal{R}$ a simple region if, for some  polygon $\mathcal{P}$, $\mathcal{R}$ is the union of $\mathcal{P}$ and its interior.

*Two regions are congruent if their associated polygons are congruent.

*Two simple regions are said to be non-overlapping if their interiors are disjoint.

*A polygon is called a square if it has four sides, all four sides are congruent and all four angles have measure 90.

*We describe a region as a square region if its associated polygon is a square.
Postulates of Hyperbolic Geometry
Postulates for hyperbolic geometry are the postulates are Postulates 1–9 of neutral geometry, plus the following:
Postulate 10H (The Hyperbolic Parallel Postulate). For each line $\ell$ and each point $A$ that does not lie on $\ell$, there are at least two distinct lines that contain $A$ and are parallel to $\ell$.
A: Postulates used in George D. Birkhoff's A set of postulates for plane geometry, based on scale and protractor (1932) 
The undefined terms in Birkhoff's axiomatization are point, line, distance and angle.
Postulate I (The Postulate of Line Measure). The points $A, B, \dots$ of any line can be put into $1:1$ correspondence with the real numbers $x$ so that $|x_B-x_A| = d(A,B)$ for all points $A$ and $B$.
Postulate II (The Point-line Postulate). One and only one line, $\ell$, contains any two distinct points $P$ and $Q$.
Postulate III (The Postulate of Angle Measure). The half-lines, or rays, $\ell, m, \dots$ through any point $O$ can be put into $1:1$ correspondence with the real numbers $a \; ( \textrm{ mod }2 \pi)$ so that if $A \neq 0$ and $B \neq 0$ are points of $\ell$ and $m$, respectively, the difference $a_m - a_l \; ( \textrm{ mod }2 \pi)$ of the numbers associated with lines $\ell$ and $m$ is $\angle AOB$. Furthermore, if the point $B$ on $m$ varies continuously in a line $r$ not containing the vertex $O$, the number $a_m$ varies continuously also.
Postulate IV (Postulate of Similarity). If in two triangles $\triangle ABC$ and $\triangle A'B'C'$ and for some constant $k > 0,\; d(A', B') = kd(A, B),\; d(A', C') = kd(A, C)$, and $\angle B'A'C' = \pm \angle BAC$, then also $d(B', C') = kd(B, C),\; \angle C'B'A' = \pm \angle CBA$, and $\angle A'C'B' = \pm \angle ACB$.
A: Axioms used in Euclid's elements as translated by J. L. Heiberg
Link: http://farside.ph.utexas.edu/euclid/Elements.pdf
Definitions 


*

*A point is that of which there is no part. 

*And a line is a length without breadth. 

*And the extremities of a line are points. 

*A straight-line is (any) one which lies evenly with points on itself. 

*And a surface is that which has length and breadth only. 

*And the extremities of a surface are lines. 

*A plane surface is (any) one which lies evenly with the straight-lines on itself. 

*And a plane angle is the inclination of the lines to one another, when two lines in a plane meet one another, and are not lying in a straight-line.

*And when the lines containing the angle are straight then the angle is called rectilinear.

*And when a straight-line stood upon (another) straight-line makes adjacent angles (which are) equal to one another, each of the equal angles is a right-angle, and the former straight-line is called a perpendicular to that upon which it stands.

*An obtuse angle is one greater than a right-angle. 

*And an acute angle (is) one less than a right-angle. 

*A boundary is that which is the extremity of something. 

*A figure is that which is contained by some boundary or boundaries. 

*A circle is a plane figure contained by a single line [which is called a circumference], (such that) all of the straight-lines radiating towards [the circumference] from one point amongst those lying inside the figure are equal to one another.

*And the point is called the center of the circle.

*And a diameter of the circle is any straight-line, being drawn through the center, and terminated in each direction by the circumference of the circle. (And) any such (straight-line) also cuts the circle in half.

*And a semi-circle is the figure contained by the diameter and the circumference cuts off by it. And the center of the semi-circle is the same (point) as (the center of) the circle.

*Rectilinear figures are those (figures) contained by straight-lines: trilateral figures being those contained by three straight-lines, quadrilateral by four, and multi- lateral by more than four.

*And of the trilateral figures: an equilateral trian- gle is that having three equal sides, an isosceles (triangle) that having only two equal sides, and a scalene (triangle) that having three unequal sides.

*And further of the trilateral figures: a right-angled triangle is that having a right-angle, an obtuse-angled (triangle) that having an obtuse angle, and an acute- angled (triangle) that having three acute angles.

*And of the quadrilateral figures: a square is that which is right-angled and equilateral, a rectangle that which is right-angled but not equilateral, a rhombus that which is equilateral but not right-angled, and a rhomboid that having opposite sides and angles equal to one an- other which is neither right-angled nor equilateral. And let quadrilateral figures besides these be called trapezia.

*Parallel lines are straight-lines which, being in the same plane, and being produced to infinity in each direc- tion, meet with one another in neither (of these direc- tions).


Postulates


*

*Let it have been postulated† to draw a straight-line from any point to any point.

*And to produce a finite straight-line continuously in a straight-line.

*And to draw a circle with any center and radius. 

*And that all right-angles are equal to one another. 

*And that if a straight-line falling across two (other) straight-lines makes internal angles on the same side (of itself whose sum is) less than two right-angles, then the two (other) straight-lines, being produced to infinity, meet on that side (of the original straight-line) that the (sum of the internal angles) is less than two right-angles (and do not meet on the other side).


Common Notions


*

*Things equal to the same thing are also equal to one another.

*And if equal things are added to equal things then the wholes are equal.

*And if equal things are subtracted from equal things then the remainders are equal.

*And things coinciding with one another are equal to one another.

*And the whole [is] greater than the part.

