  We study asymptotics of fiber integrals depending on a large parameter. When
the critical fiber is singular, full-asymptotic expansions are established in
two different cases : local extremum and isolated real principal type
singularities. The main coefficients are computed and invariantly expressed. In
the most singular cases it is shown that the leading term of the expansion is
related to invariant measures on the spherical blow-up of the singularity. The
results can be applied to certain degenerate oscillatory integrals which occur
in spectral analysis and quantum mechanics.
