Interacting galaxies show a variety of exotic morphologies besides the well-known rings, spirals, and bars. It is proposed that many of these can be identified with nonlinear, caustic structures that are known from singularity theory to be generic in collisionless dynamics. A simple kinematical model for stellar orbits is used to study the development of caustic waveforms in galaxy disks following impulsive, nonaxially symmetric (collisional) disturbances. It appears that all of the caustics known from singularity theory to be generic in two dimensions develop as transient forms in at least some moderately offcenter collisions. The more complex, higher-order caustics appear most prominently in the more offcenter and larger amplitude disturbances. It is further suggested that the highly nonlinear caustic waves may play an important role in interaction-induced star formation.