As noted above, the collection of all sets cannot be a set in any set theory with specification, regardless of regularity:

Call the collection of all sets S. We specify T by

T=\{x| x \in S \wedge \neg x \in x\}

T is then the subclass of S of sets that do not contain themselves. Thus with specifiction (provable from replacement, or as an axiom by itself), it is contradictory for S to be a set.

If you reject the law of excluded middle, you can have a intuitionist set theory where S is neither a set nor not a set; alternatively, you can have a set theory without specification one constructed based on a type theory might meet this requirement.