Abstract: Consider a solution to the semiclassical Schr\"odinger equation with a potential that has a discontinuity in some high derivative. There are subtle corrections to the naive WKB Ansatz in such a situation, reflecting the fact that a small amount of energy is reflected from the derivative discontinuity. These corrections can be viewed as a form of diffraction, since they are not visible at the level of classical particle trajectories. I will discuss the propagation of semiclassical singularities in the presence of singular potentials, as well as applications to the distribution of resonant states for open systems. This is joint work with Oran Gannot.