  We consider the quantum dynamics of gravitational collapse in a model in
which the wave function spreads out over a large ensemble of geometries as
envisioned in the fuzzball proposal. We show that the probabilities of
coarse-grained observables are highly peaked around the classical black hole
values. By contrast, probabilities for finer-grained observables probing the
neighbourhood of collapsed objects are more broadly distributed and no notion
of `averaging' applies to them. This implies that the formation of fuzzballs
gives rise to distinct observational signatures that are more significant than
has hitherto been thought and may be tested against observations in the near
future. We also predict a novel kind of gravitational wave burst associated
with the spreading of the wave function in gravitational collapse.
