The energy radiation of dynamical, isolated gravitational systems is governed by nonlinear geometric wave equations. For dynamical black hole spacetimes, tails in the late-time asymptotics of solutions to these equations are closely linked to the late-time dynamics of the event horizon and the nature of singularities that may be present in the black hole interior. I will describe the relation between Huygens’ principle for the standard wave equation and the presence of inverse-polynomially decaying tails for wave equations on stationary black hole spacetimes—a phenomenon first predicted in the physics literature and now known as “Price’s law.” I will then discuss recent work with Lionor Kehrberger on departures from Price’s law when considering dynamical spacetimes or nonlinear equations. Finally, I will illustrate how late-time tails change drastically when adding charge to the wave equation and why this is connected to the presence of instabilities on extremal black holes.