The purpose of this article is to investigate the structure of singular measures from a micro-local perspective. We introduce a notion of L 1-regularity wave front set for scalar and vectorial distributions. Our main result is a proper microlocal characterisation of the support of the singular part of temperate Radon measures and of their polar functions at these points. We deduce a sharp L 1 elliptic regularity result which appears to be new even for scalar measures. We also deduce several consequences including extensions of the results in [9] giving constraints on the polar function at singular points for measures constrained by a PDE. Finally, we also illustrate the interest of this micro-local approach with a result of propagation of singularities for constrained measures.