  We give a quantitative version of Vainberg's method relating pole free
regions to propagation of singularities for black box scatterers. In
particular, we show that there is a logarithmic resonance free region near the
real axis of size $\tau$ with polynomial bounds on the resolvent if and only if
the wave propagator gains derivatives at rate $\tau$. Next we show that if
there exist singularities in the wave trace at times tending to infinity which
smooth at rate $\tau$, then there are resonances in logarithmic strips whose
width is given by $\tau$. As our main application of these results, we give
sharp bounds on the size of resonance free regions in scattering on
geometrically nontrapping manifolds with conic points. Moreover, these bounds
are generically optimal on exteriors of nontrapping polygonal domains.
