After recalling some regularity results on the discontinuity set of Mumford-Shah minimizers, we discuss an epsilon-regularity result at the endpoint of connected arcs in 2-dimensions obtained in a joint work with  C. De Lellis (U. Zuerich). As an outcome of our analysis, if in a ball  B_r(x) the jump set of a given Mumford-Shah minimizer is sufficiently close in the Hausdorff distance to a radius of B_r(x), then in a smaller ball the jump set is a connected arc terminating at some interior point and C^{1,\alpha} up to the tip.