We calculate the effects of random density fluctuations on two-neutrino flavor transformations (()e) in the post-core-bounce supernova environment. In particular, we follow numerically the flavor evolution of neutrino states propagating through a stochastic field of density fluctuations. We examine the approach to neutrino flavor depolarization and study the effects of this phenomenon in both the early shock reheating epoch and the later r-process nucleosynthesis epoch. Our results suggest that significant fluctuation-induced neutrino flavor depolarization effects occur in these environments only when the zero-order (without density fluctuations) evolution of the neutrino states includes adiabatic propagation through resonances (mass level crossings). In the shock reheating epoch, depolarization effects from fluctuations with amplitudes larger than 0.05% of the local matter density can cause an increase in the heating rate of the material behind the shock compared to the case with no neutrino flavor transformation, but this corresponds to a significant decrease in this quantity relative to the case with adiabatic neutrino transformation. If r-process nucleosynthesis is to occur during the late stages of supernova evolution, then the requirement of neutron-rich conditions excludes a region of neutrino mass-squared difference and vacuum mixing angle (m2, sin22) parameter space for neutrino flavor transformation. We find that in the presence of stochastic fluctuations, this excluded region is not significantly altered even for random fluctuations with an amplitude of 1% of the local matter density.