  We characterize the growth and spreading of operators and entanglement in two
paradigmatic non-thermalizing phases - the many-body localized phase and the
random singlet phase - using out-of-time-ordered correlators, the entanglement
contour, and operator entanglement. We contrast these phases with strongly
thermalizing holographic conformal field theories and fully localized Anderson
insulators. We obtain a phenomenological description of the operator and state
dynamics of these phases and demonstrate the utility of the entanglement
contour and operator entanglement measures as useful probes of slowly
scrambling and non-thermalizing dynamics.
