We investigate the regularity of the free boundaries in the 3 elastic membranes problem. We show that the two free boundaries corresponding to the coincidence regions between consecutive membranes are C1,log-hypersurfaces near a regular intersection point. We also study two types of singular intersections. The first type of singular points are locally covered by a C1,α-hypersurface. The second type of singular points stratify and each stratum is locally covered by a C1-manifold. In the physical dimension d=2, these fully resolve the free boundary regularity in this problem.