  This paper is an attempt to classify finite-time singularities of PDEs. Most
of the problems considered describe free-surface flows, which are easily
observed experimentally. We consider problems where the singularity occurs at a
point, and where typical scales of the solution shrink to zero as the
singularity is approached. Upon a similarity transformation, exact self-similar
behaviour is mapped to the fixed point of a {\it infinite dimensional dynamical
system} representing the original dynamics. We show that the dynamics close to
the fixed point is a useful way classifying the structure of the singularity.
Specifically, we consider various types of stable and unstable fixed points,
centre-manifold dynamics, limit cycles, and chaotic dynamics.
