
The simplest example of a curve with a singularity is typically the
semicubical parabola. This can be parametrised by intersecting the
line through the origin of slope $a$ with the curve $y^2=x^3$ to get
$(x,y) = (a^2,a^3)$.
More generally, if higher degree curves have enough singularities to make
them genus 0 then one can always parameterise them  however one may
need to use higher degree parametrising curves.
Animated parametrisations of singular cubic, quintic and septic curves show the remarkable fact that the only points of intersection of the two curves are at the nodes, cusps and precisely one extra moving pointshown in yellow. Many thanks to Michael "Immir" Smith for generating the first two cubic plots. 
Ralph Buchholz
11 June 2016