Self-similar solutions play an important role in the dynamics of nonlinear wave equations as they provide explicit examples for finite-time blowup. This talk will be concerned with the focusing cubic and the quadratic wave equation, respectively, in dimensions where the models are energy supercritical. For both equations, we present new self-similar solutions with non-trivial profiles, which are completely explicit in all supercritical dimensions. Furthermore, we analyse their stability locally in backward light cones without symmetry assumptions. This involves a delicate spectral problem that we are able to solve rigorously only in particular space dimensions. In these cases, we prove that the solutions are co-dimension one stable modulo translations in a backward light cone of the blowup point. This is joint work with Irfan Glogić (Vienna) and Elek Csobo (Innsbruck).