We survey rigorous, formal, and numerical results on the formation of point-like singularities (or blow-up) for a wide range of evolution equations. We use a similarity transformation of the original equation with respect to the blow-up point, such that self-similar behaviour is mapped to the fixed point of an infinite dimensional dynamical system. We point out that analysing the dynamics close to the fixed point is a useful way of classifying the structure of the singularity. As far as we are aware, examples from the literature either correspond to stable fixed points, low-dimensional centre-manifold dynamics, limit cycles, or travelling waves. We will point out unsolved problems, present perspectives, and try to look at the role of geometry in singularity formation.