Slope of a line segment. If $A(x_1, y_1)$ and $B(x_2, y_2)$, we know that slope $m = \frac {(y_2 - y_1)} {(x_2 - x_1)}$.
What decision can we take aout the line segment when, $m = \frac 0 0$, $m = \frac {dy} 0$, and, $m = \frac 0{dx}$? 
If $A(0,0)$ and $B(0, 10)$, what would be the slope of the line?
 A: This formula is true if $A\neq B$. Then, in the first case you are taking $A=B$ and the formula does not work. When $x_1=x_2$ for all $A\neq B$ the slope is not defined (or it can be regarded as infinite). Last case, we have a horizontal line.
A: A horizontal line segment (one with a zero change in the $y$-coordinates but a non-zero change in the $x$-coordinates) is said to have a slope of $0$, since dividing $0$ by anything non-zero gives you zero.
A line segment with a zero change in the $x$-coordinates is always said to have an undefined slope, since the answer to anything divided by $0$ is always "undefined".
It is sometimes stated, however, that a vertical line segment (one with a zero change in the $x$-coordinates but a non-zero change in the $y$-coordinates) is said to have an "infinite slope".  This is not to say, however, that $1/0 = \infty$; division by zero is always "undefined".
If both the top and the bottom are zero, then we can't say anything about the slope of the "line segment" since we only have one point.
A: I wish you use different notation; $dx$ is for infinitesmals, $\Delta x $ for small and finite segments of a line. Anyway,
$m = \frac 0 0$ is indeterminate direction, $m = \frac {dy} 0$ is for a line parallel to y-axis, and  $m = \frac 0{dx}$ is for a line parallel to x-axis. 
A: Please be careful not to use $dx$ and $dy$ to mean the difference between the $x$ and $y$ coordinates. It's better to use $\Delta x$ and $\Delta y$ for these differences. (The Greek letter $\Delta$ is pronounced "delta".)
If you have two distinct points, say $P(x_1,y_1)$ and $Q(x_2,y_2)$ then there is a unique line passing through $P$ and $Q$. As you correctly say: the gradient of this line is given by
$$m = \frac{y_1-y_2}{x_1-x_2} = \frac{\Delta y}{\Delta x}$$
If $\Delta x = 0$ and $\Delta y = 0$ then $x_1=x_2$ and $y_1=y_2$. This means $P=Q$ and so $P$ and $Q$ are not distinct and there are many lines passing through a single point, and no single gradient.
If $\Delta x = 0$ and $\Delta y \neq 0$ then $x_1=x_2$. This means that $P$ and $Q$ lie directly above/below one another, i.e. the line through $P$ and $Q$ is a vertical line. If $x_1=x_2=k$ then the equation of this line will be $x=k$. If you want to be naughty then we could say that the gradient is infinite.
If $\Delta x \neq 0$ and $\Delta y = 0$ then $y_1=y_2$. This means that $P$ and $Q$ lie directly to the left/right of one another, i.e. the line through $P$ and $Q$ is a horizontal line. If $y_1=y_2=k$, then the equation of this line will be $y=k$. In this case the gradient is zero.
In the case $A(0,0)$ and $B(0,10)$ then $\Delta x = 0$ and $\Delta y \neq 0$. The equation of the line through $A$ and $B$ will be $x=0$, i.e. the $y$-axis.
