  Sparsity properties for phase-space representations of several types of
operators have been extensively studied in recent papers, including
pseudodifferential, Fourier integral and metaplectic operators, with
applications to time-frequency analysis of Schr\"odinger-type evolution
equations. It has been proved that such operators are approximately
diagonalized by Gabor wave packets. While the latter are expected to undergo
some spreading phenomenon, there is no record of this issue in the
aforementioned results. In this paper we prove refined estimates for the Gabor
matrix of metaplectic operators, also of generalized type, where sparsity,
spreading and dispersive properties are all noticeable. We provide applications
to the propagation of singularities for the Schr\"odinger equation.
