Abstract: We study the development of caustics in shift-symmetric scalar field theories by focusing on simple waves with an SO(p)-symmetry in an arbitrary number of space dimensions. We show that the pure Galileon, the DBI-Galileon, and the extreme-relativistic Galileon naturally emerge as the unique set of caustic-free theories, highlighting a link between the caustic-free condition for simple SO(p)-waves and the existence of either a global Galilean symmetry or a global (extreme-) relativistic Galilean symmetry.