The classical obstacle problem consists of finding the equilibrium position of an elastic membrane whose boundary is held fixed and which is constrained to lie above a given obstacle. By classical results of Caffarelli, the free boundary is $C^\infty$ outside a set of singular points. Explicit examples show that the singular set could be in general (n-1)-dimensional \emdash that is, as large as the regular set. In a recent paper with Ros-Oton and Serra we show that, generically, the singular set has zero $H^{n-4}$ measure (in particular, it has codimension three inside the free boundary), solving a conjecture of Schaeffer in dimension $n \leq 4$. The aim of this talk is to give an overview of these results.