  The aim of this note is to review some recent developments on the regularity
theory for the stationary and parabolic obstacle problems. After a general
overview, we present some recent results on the structure of singular free
boundary points. Then, we show some selected applications to the generic
smoothness of the free boundary in the stationary obstacle problem (Schaeffer's
conjecture), and to the smoothness of the free boundary in the one-phase Stefan
problem for almost every time.
