In mathematics, a **singularity** is in general a point at which a given mathematical object is not defined or lacks some "nice" property, such as differentiability.

For example, the function *f*(*x*) = 1/*x* has a singularity at *x* = 0, where it explodes to ±∞ and isn't defined. The function *g*(*x*) = |*x*| (see absolute value) also has a singularity at *x* = 0, since it isn't differentiable there. The algebraic set defined by *y*^{2} = *x*^{2} in the (*x*,*y*) coordinate system has a singularity at (0,0) because it doesn't admit a tangent there. The algebraic set defined by *y*^{2} = *x* also has a singularity at (0,0), this time because it has a "corner" at that point.

In complex analysis, we distinguish three types of singularities. Suppose *U* is an open subset of **C**, *a* is an element of *U* and *f* is a holomorphic function defined on *U*-{*a*}.

- the point
*a*is a removable singularity of*f*if there exists a holomorphic function*g*defined on all of*U*such that*f*(*z*)=*g*(*z*) for all*z*in*U*-{*a*}. - the point
*a*is a pole of*f*if there exists a holomorphic function*g*defined on*U*and a natural number*n*such that*f*(*z*) =*g*(*z*) / (*z*-*a*)^{n}for all*z*in*U*-{*a*}. - the point
*a*is an essential singularity of*f*if it is neither a removable singularity nor a pole.