The study of regularity in free probability boils down to the question of how much information about a *-algebra can be gleaned from probabilistic properties of its generators. Some of the first results in this theme come from the theory of Voiculescu's free entropy: generators satisfying certain entropic assumptions generate von Neumann algebras which are non-Gamma, or prime, or do not admit Cartan subalgebras. Free Stein dimension -- a quantity I introduced with Nelson -- is a more recent quantity in a similar vein, which is robust under polynomial transformations and not trivial for variables which do not embeddable in R^\omega.

In this talk, I will recall the motivation and definition of free Stein dimension, and spend some time focusing on how (approximate) algebraic relations between generators can be used to provide upper bounds on the Stein dimension; of particular interest are commutation, and good behaviour under conjugation, and I will mention how these results apply in some interesting examples. Time permitting I will discuss how free Stein dimension behaves under ``building block'' operations such as direct sums and tensor products with finite dimensional algebras. This is joint work with Brent Nelson.