The obstacle problem arises in several physical models: from electrostatics to fluid filtration. I will present some recent work in collaboration with A. Figalli and X. Ros-Oton on the structure of the singular set for this problem and the generic regularity of the free boundary.

By using new methods ---based on an existing connection between the obstacle problem and the thin obstacle problem--- we can give a very accurate description of the behaviour of solutions to the obstacle problem near "generic" singularities.

As a consequence we are able to use a Sard type approach in order to show that the free boundary is generically smooth in dimensions 3 and 4, while in higher dimensions we prove that the singular set has, generically, co-dimension 4.