According to a result of Higman and Sims (which I learned about in this paper of Poonen's) the typical p-group is 3-step nilpotent of a particular form. In particular the typical group is a 3-step nilpotent 2-group of a particular form. By typical here I mean that eventually the number of these groups dominate.
Is anything known about what the typical non-solvable group looks like? Probably some sort of modification of PSL_2(F_p)?