The focus of this talk is new results on the partial regularity of the free-boundary for local minimizes of a class of degenerate Alt-Caffarelli functionals. These degenerate functionals arise naturally in variational formulations of the Stokes wave problem. But, because of their degeneracy, the standard tools for proving weak geometric regularity (interior ball conditions) leave open the possibility that cusps may form. Recently, this question was answered in joint work with Lisa Naples.