Two-dimensional unit-length vector fields of vanishing divergence

We prove the following regularity result: any two-dimensional unit-length divergence-free vector field belonging to W1/p,p (p∈[1,2]) is locally Lipschitz except at a locally finite number of vortex-point singularities. We also prove approximation results for such vector fields: the dense sets are formed either by unit-length divergence-free vector fields that are smooth except at a finite number of points and the approximation result holds in the -topology (1⩽q<2), or by everywhere smooth unit-length vector fields (not necessarily divergence-free) and the approximation result holds in a weaker topology.