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We show that certain singular structures (H\"{o}lderian cusps and milddivergences) are transported by the flow of homeomorphisms generated by anOsgood velocity field. The structure of these singularities is related to themodulus of continuity of the velocity and the results are shown to be sharp inthe sense that slightly more singular structures cannot generally bepropagated. For the 2D Euler equation, we prove that certain singularstructures 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 anonlinear fashion, up to bounded perturbations. We also give stability resultsfor weak Euler solutions away from their singular set.

Propagation of singularities by Osgood vector fields and for 2D inviscid incompressible fluids