The authors discuss smoothing of initial states under the time development of the Schrödinger equation and generalizations thereof. They show that the microlocal smoothness of a solution at a point in phase space depends on non-trapping conditions on the bicharacteristic flow and the microlocalized moment properties of the initial state. This generalizes the fact that the smoothness of a solution of the free Schrödinger equation depends on the moments of the initial states which can be checked directly. These results contrast to analogous results for the propagation of singularities for hyperbolic equations.