  We show that certain singular structures (H\"{o}lderian cusps and mild
divergences) are transported by the flow of homeomorphisms generated by an
Osgood velocity field. The structure of these singularities is related to the
modulus of continuity of the velocity and the results are shown to be sharp in
the sense that slightly more singular structures cannot generally be
propagated. For the 2D Euler equation, we prove that certain singular
structures are preserved by the motion, e.g. a system of $\log\log_+(1/|x|)$
vortices (and those that are slightly less singular) travel with the fluid in a
nonlinear fashion, up to bounded perturbations. We also give stability results
for weak Euler solutions away from their singular set.

Analysis of PDEs;Classical Analysis and ODEs;Fluid Dynamics