  We make the first testable predictions for the local two-particle spectral
function of a disordered s-wave superconductor, probed by scanning Josephson
spectroscopy (sjs), providing complementary information to scanning tunneling
spectroscopy (sts). We show that sjs provides a direct map of the local
superconducting order parameter that is found to be anticorrelated with the gap
map obtained by sts. Furthermore, this anticorrelation increases with disorder.
For the momentum resolved spectral function, we find the Higgs mode shows a
non-dispersive subgap feature at low momenta, spectrally separated from phase
modes, for all disorder strengths. The amplitude-phase mixing remains small at
low momenta even when disorder is large. Remarkably, even for large disorder
and high momenta, the amplitude-phase mixing oscillates rapidly in frequency
and hence do not affect significantly the purity of the Higgs and phase
dominated response functions.
