  We review some classical results and more recent insights about the
regularity theory for local minimizers of the Mumford and Shah energy and their
connections with the Mumford and Shah conjecture. We discuss in details the
links among the latter, the porosity of the jump set and the higher
integrability of the approximate gradient. In particular, higher integrability
turns out to be related with an explicit estimate on the Hausdorff dimension of
the singular set and an energetic characterization of the conjecture itself.
