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

Fine regularity results for Mumford-Shah minimizers: porosity, higher integrability and the Mumford-Shah conjecture