Consider $f=x$ and $g=x^2$ as elements of $\mathbb R[x,y]$, which define the same curve, namely the $y$-axis. 

According to the usual definition of a singular point (vanishing partial derivatives), it then seems to depend on the given polynomial whether a point on that curve is singular, although we are dealing with the same object. For $f$, all points are non-singular whereas they are all singular for $g$. I've also seen a definition that requires to consider the generators of the ideal of the curve, in this case it would be the ideal $(x)$ and therefore the curve non-singular. For me, this definition seems to be "more" correct because our curve doesn't look singular at any point. How do you handle this "problem"?