Chaotic Hamiltonians famously display random matrix universality in their spectral statistics. While it is still not known how to explain this universality via a renormalization group treatment, it is possible to define coarse-graining procedures whose fixed points diagnose quantum ergodicity.  Systematically applying this tool to various theories leads to appealing phenomenological rules that link quantum chaos with irregular singular structures in phase space.  In this talk I will describe these ideas and argue that they present a unified frame that spans concepts as diverse as black holes, spin glasses, and many-body-localized phases of matter.