  Scrambling is a key concept in the analysis of nonequilibrium properties of
quantum many-body systems. Most studies focus on its characterization via
out-of-time-ordered correlation functions (OTOCs), particularly through the
early-time decay of the OTOC. However, scrambling is a complex process which
involves operator spreading and operator entanglement, and a full
characterization requires one to access more refined information on the
operator dynamics at several timescales. In this work we analyze operator
scrambling by expanding the target operator in a complete basis and studying
the structure of the expansion coefficients treated as a coarse-grained
probability distribution in the space of operators. We study different features
of this distribution, such as its mean, variance, and participation ratio, for
the Ising model with longitudinal and transverse fields, kicked collective spin
models, and random circuit models. We show that the long-time properties of the
operator distribution display common features across these cases, and discuss
how these properties can be used as a proxy for the onset of quantum chaos.
Finally, we discuss the connection with OTOCs and analyze the cost of probing
the operator distribution experimentally using these correlation functions.
