We discuss an epsilon-regularity result at the endpoint of connected arcs for 2-dimensional Mumford-Shah minimizers obtained in a joint work with C. De Lellis (U. Zuerich). As an outcome of our analysis, if in a ball Br(x) the jump set of a given Mumford-Shah minimizer is sufficiently close in the Hausdorff distance to a radius of Br(x), then in a smaller ball the jump set is a connected arc terminating at some interior point and C1,α up to the tip.