Richards equation is ubiquitous in the modelling of flows in porous media. It serves as a model in its own right, but also as a stepping stone to more complex models of multiphase flows. Despite its relative simplicity, it provides many challenges from a computational point of view due to the nonsmooth and degenerate nature of the functions that appear in the equation. In this paper, we replace these functions with regularized (smooth and nondegenerate) counterparts where the amount of added regualrization is controlled by a single parameter. We also introduce a set of a posteriori error estimators that we use to construct adaptive stopping criteria. In particular, we use a stopping criterion to reduce the decrease the regularization parameter, which we refer to as adaptive regularization. The full adaptive algorithm is tested on a suite of numerical examples adapted from other recent works on improving the robustness of solvers for the Richards equation.