In this talk, I will discuss the relationship between minimal surfaces and the Bernoulli one-phase free boundary problem through capillarity. We construct non-flat minimal capillary cones with any contact angle, notably including minimizing capillary and one-phase examples with bi-orthgonal symmetry. These cones interpolate between rescalings of a singular solution to the one-phase problem and the free-boundary minimal surfaces. These methods produce new minimal and CMC surfaces in the sphere and one-phase cones with intricate topologies. We further show regularity results proving certain singular capillary cones are minimizing in ambient dimension 8 or higher, demonstrating that the regularity theory for minimizing capillary hypersurfaces can have singularities in codimension 7 and proving the optimal regularity for contact angles near $\pi/2$ as well as providing new singular minimizing solutions to the one-phase problem. This is joint work with R. Tsiamis and Y. Wang.