  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 tempered 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.
