We consider an equation in divergence form with a singular/degenerate weight. We first study the regularity of the nodal sets of solutions in the linear case. Next, when the r.h.s. does not depend on u, under suitable regularity assumptions, we prove Hölder continuity of solutions and possibly of their derivatives up to order two or more (Schauder estimates). In addition, we show stability of the C^{0,α} and C^{1,α} a priori bounds for approximating problems Finally, we derive C^{0,α} and C^{1,α} bounds for inhomogenous Neumann boundary problems as well. Our method is based upon blow-up and appropriate Liouville type theorems.