The modern theory of simple tracial C*-algebras is characterized by a rich collection of divisibility conditions, coming in numerous flavours and levels of strength. Z-stability and almost divisibility are primary examples of what might be called 'Cuntz-type' divisibility, while in the tracial setting we have counterparts like uniform property Gamma and tracial almost divisibility. On the dynamical side, analogous phenomena emerge through conditions like the small boundary property, the uniform Rokhlin property, and almost finiteness (in measure), often translating into 'relative' versions of the aforementioned properties for Cartan pairs. In this talk I will survey several of these regularity properties, alongside more recent notions introduced by Elliott and Niu, and explore some of the relationships between them. I will in furthermore try to relate some well-known instances of 'automatic centrality' which arises both in the setting of C*-algebras and in topological dynamics, by which I mean results that, under strong nuclearity assumptions, allow to upgrade non-central tracial divisibility conditions (e.g. tracial almost divisibility, small boundary property) to approximately central ones (uniform property Gamma, almost finiteness in measure).