  We study the effect of disorder on the propagation of collective excitations
in a disordered Bose superfluid. We incorporate local density depletion induced
by strong disorder at the meanfield level, and formulate the transport of the
excitations in terms of a screened scattering problem. We show that the
competition of disorder, screening, and density depletion induces a strongly
non-monotonic energy dependence of the disorder parameter. In three dimensions,
it results in a rich localization diagram with four different classes of
mobility spectra, characterized by either no or up to three mobility edges.
Implications on experiments with disordered ultracold atoms are discussed.
