For simplicity, let us consider curves in the Euclidean plane ${\mathbb{R}}^2$. They may be defined as the set of points $(x_1, x_2)$ satisfying an equation of the form $P(x_1,x_2)=0$ where $P(x_1, x_2)$ is the polynomial
${\displaystyle P(x_1, x_2)=a_{0}+b_{0}x_1+b_{1}x_2+c_{0}x_1^{2}+2c_{1}x_1x_2+c_{2}x_2^{2}+\dots }$
Definition. The singular points are those on the curve whose partial derivatives vanish $${\displaystyle \nabla P(x_1,x_2)=0 \quad \operatorname{or}\quad {\frac {\partial P}{\partial x_1}}={\frac {\partial P}{\partial x_2}}=0.}$$
I think I don't see the point (no pun intended). Usually, the term 'singularity' in maths means 'lacking of differentiability' or '(locally) bad behaviour'. The null gradient sounds more like a stationary point here. I mean.. what could go wrong if the curve has a null gradient? What's 'singular' about it?