Given a finitely supported probability measure on a Kleinian group, Kaimanovich showed that the Poisson boundary of the associated random walk is the boundary of hyperbolic space equipped with the hitting measure, which is the unique stationary measure. It has been conjectured that this stationary measure is singular with respect to conformal measures of the Kleinian group (or Patterson--Sullivan measures). In this talk, I will explain how Mostow rigidity can be generalized to show this expected singularity when the Kleinian group is not convex cocompact. If time permits, I will also discuss corresponding results for random walks on mapping class groups and stationary measures on the Thurston boundary of Teichmüller space. This is joint work with Andrew Zimmer.