  Consider in the phase space of classical mechanics a Radon measure that is a
probability density carried by the graph of a Lipschitz continuous (or even
less regular) vector field. We study the structure of the push-forward of such
a measure by a Hamiltonian flow. In particular, we provide an estimate on the
number of folds in the support of the transported measure that is the image of
the initial graph by the flow. We also study in detail the type of
singularities in the projection of the transported measure in configuration
space (averaging out the momentum variable). We study the conditions under
which this projected measure can have atoms, and give an example in which the
projected measure is singular with respect to the Lebesgue measure and diffuse.
We discuss applications of our results to the classical limit of the
Schr\"{o}dinger equation. Finally we present various examples and
counterexamples showing that our results are sharp.
