The
Zermelo-Fraenkel set theory axioms. This
formulation of
set theory is the most popular among
mathematicians. Historically it carries little "
metaphysical baggage", which might be a contributing factor to its popularity.
The axiom of choice is not included in the ZF axioms; ZF+choice is usually abbreviated ZFC.
While all of mathematics is supposed to be embedded in ZFC (or just ZF), some mathematicians prefer to work outside it (or to pretend to work outside it, depending on whom you ask). Some logicians use other systems (which are essentially equivalent); some category theorists prefer to avoid using standard formulations of set theory, since they need to use many large categories. And Conway has advocated (in On Numbers and Games) ditching the entire fixation on axiom systems, and returning to a more relaxed view of the foundations of mathematics wherever possible (he also uses very large objects, such as a category which satisfies the field axioms, except for not being a set).