Peter, there is a nice paper on precisely this topic. It is not quite exhaustive, but it provides an extensive bibliography and a good overview:
M. Randall Holmes, Thomas Forster and Thierry Libert. Alternative Set Theories. In Handbook of the History of Logic, vol. 6, “Sets and Extensions in the Twentieth Century”, 2012, Elsevier/North-Holland, Dov Gabbay, Akihiro Kanamori, and John Woods, eds., pp. 559-632.
For some reason, the volume does not appear on MathSciNet, so I'll say a few words about the contents of the paper.
The paper begins by discussing (Section 2) Simple Type Theory, Mac Lane's and Zermelo's, not because they are alternative theories, but because they are needed to understand some of them (such as New Foundations and its variants), though the authors mention that "Zermelo set theory or variants of Zermelo set theory have been pressed into service themselves as alternative set theories, presumably by workers nervous about the high consistency strength of $\mathsf{ZFC}$."
Section 3 covers theories with classes: First Von Neumann-Gödel-Bernays and Kelley-Morse set theory, then Ackermann set theory (where non-set classes can belong to other classes, this theory is equiconsistent with $\mathsf{ZF}$), and then a weak system that they call "Pocket set theory", an expansion due to Holmes of a suggestion by Rudy Rucker.
Section 4 covers theories with atoms and with anti-foundation axioms. They first discuss $\mathsf{ZFA}$, then Aczel's anti-foundation axiom, and Boffa's axiom (in this system, there is a proper class of $x$ with $x=\{x\}$, while in Aczel's system there is only one).
They continue in section 5 with New Foundations and related systems (such as the much better understood $\mathsf{NFU}$, where urelements are allowed). Naturally, this section occupies the main bulk of the paper.
Section 6 discussed Positive set theory and its fragments and variants, typically denoted $\mathsf{PST}$, perhaps with sub- and superscripts (an exception to this notation is the theory $\mathsf{GPK}^+_\infty$, mutually interpretable with an extension of Kelley-Morse by large cardinals). This leads to Topological set theory.
Since the systems in section 6 allow talk of super- or hyperuniveses, section 7 discusses systems motivated by non-standard analysis, such as Nelson's Internal set theory, or Vopěnka's.
Section 8 concludes the list and covers "curiosities": The double extension set theory of Andrzej Kisielewicz (that "has the property which is usually ascribed to New Foundations (we believe not entirely fairly) of being motivated by a syntactical trick without any semantic motivation"), and Zermelo's set theory extended by an axiom asserting that there is an elementary $j:V\to V$, which turns out to be significantly high in terms of consistency strength
Let me add that the theory $\mathsf{ZF}$ augmented by such an axiom has also been studied, even fairly recently, but I would not classify it by any means as alternative. In general, extensions of $\mathsf{ZF}$ via large cardinals, forcing, or inner-model theoretic considerations are just part of the standard set theoretic landscape.
Something that the paper definitely does not cover is systems motivated by algebraic geometry, topos theoretic, or categorical considerations, such as Grothendieck universes. On the topic of categorical set theory, and Lawvere’s Elementary Theory of the Category of Sets, there is a recent paper that has gathered some attention,
Tom Leinster. Rethinking set theory. ArXiv:1212.6543.
Finally, there is also Bourbaki's set theory, about which Mathias has written a few critiques. You may enjoy the discussion at the nForum.
