I will sketch a proof that if Γ↷X is a continuous action of a countable group on a compact Polish space, then there is a Γ-invariant Borel set B⊆X such that Γ↷B does not have a σ-compact model. The primary new technical tool is a result asserting the generic inexistence of stationary probability measures. This is joint work with Clinton Conley and Alexander Kechris.