  We address the open problem of existence of singularities for the complex
Ginzburg-Landau equation. Using a combination of rigourous results and
numerical computations, we describe a countable family of self-similar
singularities. Our analysis includes the super-critical non-linear Schroedinger
equation as a special case, and most of the described singularities are new
even in that situation. We also consider the problem of stability of these
singularities.
