Examples of smooth curves of genus $0$ and degree $d>2$. Can we provide a source of explicit examples ?
The degree assumption $d>2$ means that I would like to see examples which are not conics.
 A: The arithmetic genus of a curve $C\subset \mathbb P^2_k$ of degree $d$ is the number $p_a(C)=dim_k H^1(C, \mathcal O_C)$ and is equal to $\frac {(d-1)(d-2)}{2}$.
Thus that arithmetic genus is $\gt0$ for curves of degree $d\gt2$.
However that curve $C$ has a geometric genus too, equal to the arithmetic genus of the normalization $\tilde C$ : $p_g(C)=p_a(\tilde C)$.
Of course these genera coincide for normal (=nonsingular)  curves .  
The geometric genus may be zero for curves of degree $d\gt0$ : for example the cusp $C$ given by $y^2z=x^3$ in $\mathbb P^2$ has geometric genus $0$, since its normalization is the projective line $\mathbb P^1$:$$\mathbb P^1=\tilde C\to C:(u:v) \mapsto (x=uv^2:y=v^3: z=u^3)      $$  
Edit: Addressing the competition
The right honourable user 64494 in his answer below gives the interesting example of the projective plane curve $x^4+x^2yz+yz^3=0$ and reports that his mysterious friend Maple (not a registered user) has computed its geometric genus to be zero.
This is confirmed by our less mysterious friend Julius Plücker, alas deceased (1801-1868), who gave the formula (cf. Seidenberg's book p.129 )$$p_g=\frac {(d-1)(d-2)}{2}  -\sum \frac {s_i(s_i-1)}{2}                 $$ for a plane projective curve having only singularities $P_i$  of  multiplicity $s_i$,  with distinct tangents   .
The curve $x^4+x^2yz+yz^3=0$ has as only singularity $(0:1:0)$, of multiplicity $3$,  with three distinct tangents $z=0,x\pm iy=0$. 
So the displayed formula yields $$p_g=\frac {(4-1)(4-2)}{2}  - \frac {3(3-1)}{2}  =0               $$  Plücker and Maple embrace each other across the centuries...
A: A smooth projective plane curve of genus $0$ has to be either a line or a conic.
But the morphism $t \mapsto (t,t^2,t^3,\ldots,t^{d-1})$ (in affine coordinates)
embeds $\mathbb P^1$ into $\mathbb P^d$ as a smooth curve of degree $d$.  (The image
is called a rational normal curve of degree $d$.)
If you choose a generic projection from $\mathbb P^d$ to $\mathbb P^3$, the image
of a degree $d$ rational normal curve will be a rational curve of degree $d$ in $\mathbb P^3$.
