Let a class \cal F of densities be given. We draw an i.i.d. sample from a density f which may or may not be in \cal F . After every n , one must make a guess whether f \in \cal F or not. A class is almost surely discernible if there exists such a sequence of classification rules such that for any f , we make finitely many errors almost surely. In this paper several results are given that allow one to decide whether a class is almost surely discernible. For example, continuity and square integrability are not discernible, but unimodality, log-concavity, and boundedness by a given constant are.