The aim of this talk is to describe how the solution of a parabolic PDE can become non-unique after a self-similar singularity. Nonlinear parabolic PDEs with a scaling are often locally well-posed in sub-critical spaces defined by the scaling. In some critical spaces (invariant with respect to the scaling), the local well-posedness might be broken, and sometimes self-similar blow-up might generate non-uniqueness. I will mainly focus on two problems in three dimensions: the incompressible Navier-Stokes equations and the complex Ginzburg-Landau equation.