I will discuss recent work, variously with Oran Gannot, Jeff Galkowski, Mengxuan Yang, and Joey Zou, on the propagation of semiclassical singularities in the presence of a low-regularity potential $V$.  We consider structured singularities, where $V$ is smooth on either side of some hypersurface $Y$; diffractive effects are known to reflect some energy from this interface even if the classical equations of motion give existence and uniqueness for particle trajectories traversing it.  A current focus is to obtain a Gutzwiller trace formula for semiclassical Hamiltonians in this setting.