Regularity properties of C*-algebras, e.g., Z-stability and strict comparison, have played crucial roles in the Elliott classification program. The equivalences (or at least close relations) between regularity properties of very different natures were predicted by the Toms-Winter conjecture and have been a central topic in the structure theory of simple nuclear C*-algebras in the last 15 years. I will discuss ways to transport these ideas to the setting of C*-dynamical systems. More precisely, I will talk about dynamical Cuntz semigroups, dynamical strict comparison, and the notion of almost elementary C*-dynamical systems, which generalizes Kerr's almost finiteness for topological dynamical systems and can be viewed as a dynamical analogue of Z-stability. This is joint work with Bosa, Perera, and Zacharias.