The aim of this seminar is to present regularity results for the gradient of solutions to some classes of strongly singular or degenerate elliptic and parabolic problems. The elliptic problem under consideration arises as the Euler-Lagrange equation of a variational integral. The energy density associated with this integral satisfies standard p-growth and p-ellipticity conditions, for p>1, with respect to the gradient variable, but only outside a ball centered at the origin. As for the parabolic problems in question, a motivation for studying them stems from gas filtration models taking into account the initial pressure gradient. The presentation will mainly focus on the higher differentiability of solutions, both of integer and fractional order. This talk is based on joint work with Fabian Bäuerlein, Antonio Giuseppe Grimaldi and Antonia Passarelli di Napoli.