I will present a construction of solutions of the Navier-Stokes equations for data whose vorticity is concentrated on 1D curves (as measures). This corresponds to large locally self-similar data, for which the usual perturbative approach to local well-posedness does not apply, and for which a number of interesting questions arise. These data are also of fundamental importance from a physical perspective, since vortex filaments are expected to play a crucial role in 3D flows. This is joint work with J. Bedrossian and B. Harrop-Griffiths.