The Mumford-Shah functional was introduced by Mumford and Shah in 1989 as a variational model for image reconstruction. The most important regularity problem is the so-called Mumford-Shah conjecture, which states that (in 2 dimensions) the closure of the jump set can be described as the union of a locally finite collection of injective $C^1$ arcs that can meet only at the endpoints, in which case they have to form triple junctions. If a point is an endpoint of one (and only one) of such arcs, it is called a cracktip. In this talk, I plan to survey some older results concerning the regularity of Mumford-Shah minimizers and their singular sets, and discuss more recent developments (joint work with Camillo De Lellis and Matteo Focardi).