The von Neumann-Day problem has two parts: Is every amenable group elementary amenable? Is every group not containing a nonabelian free group (so-called small groups) amenable? Both problems have been settled in the negative. In this talk, I will present joint work with S. Kunnawalkam Elayavalli where we show that both problems have a generic negative solution. More specifically, we present natural Polish spaces of countable amenable groups and countable small groups and prove that the set of nonamenable small groups is comeager in the space of small groups and the set of non-elementary amenable groups is comeager in the space of amenable groups. We also discuss the analogous problem for groups satisfying laws and relate it to the well-known open question of whether or not every amenable group satisfying a nontrivial law is uniformly amenable. Time permitting, we will also discuss the question of when an amenable group can have the same first-order theory as a nonamenable group.