I've recently finished my first course in axiomatic set theory, which was taught using Paul Halmos' Naive Set Theory. In it, he lays out all the axioms of ZFC set theory except for the Axiom of Regularity. From the reading I've done, it looks like it helps prevent things like Russel's Paradox, so why doesn't he include it?
First, adding an axiom cannot possibly prevent a paradox. This is because if a paradox occurs without an axiom, exactly the same paradox occurs with the axiom added in. It is only by refusing to add axioms that we can prevent paradoxes.
Second, there are very few mathematical applications of the axiom of foundation/regularity. In fact, in the presence of the axiom of choice (or in the presence of a weakened version known as the "axiom of well-founded materialisation"), there is a very precise sense in which adding the axiom of foundation makes no mathematical difference.
"Structural mathematics" is a working philosophy of mathematics in which one focuses on mathematical structures and isomorphism-invariant properties of these structures. One way of formalising "structural mathematics" is that structural mathematics is the study of statements about the category of sets.
However, assuming the axiom of choice, one can show that the category of well-founded sets is equivalent to the category of sets. Therefore, the two categories have exactly the same properties. So restricting our attention to the well-founded sets (which is essentially what the axiom of foundation does) changes nothing about structural mathematics.