A singular point of an algebraic curve is a point where the curve has "nasty" behavior such as a cusp or a point of self-intersection (when the underlying field K is taken as the reals). More formally, a point (a, b) on a curve f(x, y) = 0 is singular if the x and y partial derivatives of f are both zero at the point (a, b). (If the field K is not the reals or complex numbers, then the partial derivative is computed formally using the usual rules of calculus.) The following table gives some representative named curves that have various types of singular points at their origin.

algebraic curve | cusp | irregular singularity | ordinary point | regular singular point | singularity