The earthquake flow and the Teichmüller horocycle flow are flows on bundles over the Riemann moduli space of a surface, and are similar in many respects to unipotent flows on homogeneous spaces of Lie groups. In analogy with results of Margulis, Dani and others in the homogeneous space setting, we prove strong nondivergence results for these flows. This extends previous work of Veech. As corollaries we obtain that every closed invariant set for the earthquake (resp. Teichmüller horocycle) flow contains a minimal set, and that almost every quadratic differential on a Teichmüller horocycle orbit has a uniquely ergodic vertical foliation.