  The purpose of this article is to investigate the structure of singular
measures from a microlocal perspective. Motivated by the result of De
Philippis-Rindler [10] and the notions of wave cone of Murat-Tartar
[19,20,26,27] and of polarisation set of Denker [9] we introduce a notion of
$L^1$-regularity wave front set for scalar and vector 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.
The proof is based on De Philippis-Rindler's approach reinforced by microlocal
analysis techniques and some extra geometric measure theory arguments. We
deduce a sharp $L^1$ elliptic regularity result which appears to be new even
for scalar measures and which enlightens the interest of the techniques from
geometric measure theory to the study of harmonic analysis questions. For
instance we prove that $ \Psi^0 L^1\cap \mathcal M_{loc}\subseteq L^1_{loc},$
and in particular we obtain $L^1$ elliptic regularity results as $\Delta u\in
L^1_{loc}, D^2 u \in \mathcal M_{loc} \Longrightarrow D^2 u\in L^1_{loc}.$ We
also deduce several consequences including extensions of the results in [10]
giving constraints on the polar function at singular points for measures
constrained by a PDE, and of Alberti's rank one theorem. Finally, we also
illustrate the interest of this microlocal approach with a result of
propagation of singularities for constrained measures.
