The Stefan problem, dating back to the XIX century, aims to describe the evolution of a solid-liquid interface, typically a block of ice melting in water. A celebrated work of Luis Caffarelli from the 1970's established that the ice-water interface must be an analytic surface outside of a certain closed set: the so-called singular set. This singular set was only known to be contained in a surface of class C1 and it could be, a priori, as large as the regular set. I will present a recent joint work with A. Figalli and X. Ros-Oton in which we obtain new delicate bounds on the size (Hausdorff dimension) of the singular set in the Stefan problem.