  Our main goal is the comparative study of singularities of solutions to the
systems of first order quasilinear PDEs and their perturbations containing
higher derivatives. The study is focused on the subclass of Hamiltonian PDEs
with one spatial dimension. For the systems of order one or two we describe the
local structure of singularities of a generic solution to the unperturbed
system near the point of "gradient catastrophe" in terms of standard objects of
the classical singularity theory; we argue that their perturbed companions must
be given by certain special solutions of Painleve' equations and their
generalizations.
