What axiomatizations exist for special relativity? Einstein 1905 gives the following axiomatization of special relativity (Perrett and Jeffery's translation):


*

*The laws by which the states of physical systems undergo change 
are not affected, whether these changes of state be referred to the one or the other of two systems of coordinates in uniform translatory
motion.

*Any ray of light moves in the "stationary" system of coordinates
with the determined velocity $c$, whether the ray be emitted by a
stationary or by a moving body.
What other axiomatizations are possible?
References
Einstein, "On the electrodynamics of moving bodies," 1905, Annalen der Physik. 17 (1905) 891; English translation by Perrett and Jeffery available at http://fourmilab.ch/etexts/einstein/specrel/www/
 A: Sometimes people telescope Einstein's two postulates into one, saying that all the laws of physics (both mechanical and optical) are the same in all frames.
Einstein's axiomatization is not one that any modern physicist would choose, except for historical reasons. It represents a century-old point of view on the laws of nature, according to which there is something special about light, $c$ is the speed of light, and special relativity is a theory that has a lot to do with light. From the modern point of view, there is nothing special about light (it is just one of the massless particles in the standard model), $c$ is not really the speed of light (it's a conversion factor between space and time), and special relativity is a theory of time and space, not a theory about light. These are good reasons to prefer a different axiomatization.
Ignatowsky 1911, shortly after Einstein, did an axiomatization in terms of the symmetries of time and space. The Ignatowsky paper seems to be difficult to obtain today, and I've never found an English translation. However, a modern development along these lines is available in Pal 2003. My own presentation is here.
Einstein 1905 focuses on the problem of converting from one set of coordinates to another set based on a frame in motion relative to the original one. But one of the big lessons of general relativity was that coordinates are completely arbitrary, and for similar reasons most modern mathematicians prefer to describe noneuclidean geometry in a coordinate-independent language. Laurent 1994 is unique, as far as I know, among introductory texts on SR in that it uses this coordinate-independent approach. Laurent's central postulate is that test particles move along paths that strictly maximize proper time. The use of a strict inequality rules out Galilean relativity. He also explicitly makes the assumption that parallel transport is path-independent, which is equivalent to saying that spacetime is flat. There are some other axioms which are algebraic in character, e.g., the existence of a scalar product.
References
Einstein, "On the electrodynamics of moving bodies," 1905, Annalen der Physik. 17 (1905) 891; English translation by Perrett and Jeffery available at http://fourmilab.ch/etexts/einstein/specrel/www/
W.v. Ignatowsky, Phys. Zeits. 11 (1911) 972
Bertel Laurent, Introduction to spacetime: a first course on relativity, 1994
Palash B. Pal, "Nothing but Relativity," Eur.J.Phys.24:315-319,2003, http://arxiv.org/abs/physics/0302045v1
A: I would like to post this like a comment to the answer above, but I am not permitted to do so.
At the above answer the word "axiomatic" is used at much ambiguous ways I could say very naive at best. A real axiomatic theory must define, or assume as primitive notions, ALL the terms it uses and ALL the claims it does and that approach do not do that.
If you want to see relativity theory axiomatized in a proper way, I recommend the book by Schutz "Independent Axioms for Minkowski Space-Time" and, in a different approach, this article below (and those articles and authors which are pointed in it):
http://philsci-archive.pitt.edu/3861/1/amnszdyn-080130.pdf
I recommend some familiarity with first order logic and ideas and first definitions from model theory.
