We present recent advances in the regularity theory for weak solutions to some classes of elliptic and parabolic equations with strongly singular or degenerate structure. The equations under consideration satisfy standard $p$-growth and $p$-ellipticity conditions only outside a ball centered at the origin. In the elliptic setting, we describe Besov and Sobolev regularity results for suitable nonlinear functions of the gradient of the weak solutions, covering both the subquadratic ($1<p<2$) and superquadratic ($p\geq2$) regimes. Analogous results are obtained in the corresponding parabolic framework, where we address the higher spatial and temporal differentiability of the solutions under appropriate assumptions on the data.