  We prove an upper bound on the log canonical threshold of a hypersurface that
satisfies a certain power condition and use it to prove several generalizations
of Igusa's conjecture on exponential sums, with the log-canonical threshold in
the exponent of the estimates. We show that this covers optimally all
situations of the conjectures for non-rational singularities, by comparing the
log canonical threshold with a local notion of the motivic oscillation index.
