What combinations of realism, non-locality, and contextuality are ruled out in quantum theory? Bell's inequality theorem, along with experimental evidence, shows that we cannot have both realism and locality.  While I don't fully understand it, Leggett's inequality takes this a step further and shows that we can't even have non-local realism theories.  Apparently there are some hidden variable theories that get around this by having measurements be contextual.  I've heard there are even inequalities telling us how much quantum mechanics does or doesn't require contextuality, but I had trouble finding information on this.
This is all confusing to me, and it would be helpful if someone could explain precisely (mathematically?) what is meant by: realism, locality (I assume I understand this one), and contextuality.
What combinations of realism, locality, and contextuality can we rule out using inequality theorems (assuming we have experimental data)?
 A: This is a truly excellent question in my opinion.  It is still being worked on.
Here are some professional references that will somewhat clarify the issue, or perhaps even confuse you further:
http://arxiv.org/abs/1102.4467
http://arxiv.org/abs/1007.5518
http://arxiv.org/abs/1006.3680
Michael J.W. Hall
http://arxiv.org/abs/0808.2178
Travis Norsen
http://arxiv.org/abs/quant-ph/0209123
 Laloe, Franck
http://arxiv.org/abs/0711.4650    Adam Brandenburger, Noson Yanofsky
http://arxiv.org/abs/1102.0264   Samson Abramsky, Adam Brandenburger
A: What I take to be elementary significant papers on this question pre-date arXiv, so they are unfortunately usually available only behind paywalls. I've always found the simplicity of Willem de Muynck's argument in Physics Letters A 114, 65 (1986), "THE BELL INEQUALITIES AND THEIR IRRELEVANCE TO THE PROBLEM OF LOCALITY IN QUANTUM MECHANICS", somewhat compelling. I can reproduce the basic argument here under fair use, from the first page,

In his original derivation Bell 3
  assumed his hidden variables theory to
  satisfy a locality condition which he
  deemed to be a"vital assumption".
  Presumably due to this fact there
  still exists a widespread belief- also
  among specialists - that the Bell
  inequalities can not be derived for
  nonlocal hidden variables theories.
  This would leave open the possibility
  that quantum mechanics might be
  reproduced by a nonlocal hidden
  variables theory. From the following,
  however, it should be clear that the
  mere existence of hidden variables is
  sufficient to yield the Bell
  inequalities. Hence not only local but
  also nonlocal hidden variables
  theories are incompatible with quantum
  mechanics. Local and nonlocal theories
  being on an equal footing it also
  follows that the Bell inequalities are
  completely irrelevant to the problem
  of (non)-locality in hidden variables
  theories. (my emphasis here)

After I extracted the above, I found a PDF of the paper on de Muynck's web-page, I'm pleased to say (it's elementary math, and only 4 pages). A similar but rather more algebraic construction, which I think is mathematically quite a bit nicer, can be found in Lawrence J. Landau, Physics Letters A 120, 54, 1987, "ON THE VIOLATION OF BELL'S INEQUALITY IN QUANTUM THEORY", without, however, making anything like de Muynck's claim for its significance (I don't believe it, but I found it here). IMO, this simple algebra underlies the question of locality/nonlocality to this day -- one takes this argument seriously, or one does not.
Ultimately, locality is very closely tied to measurement compatibility because measurement compatibility is required for measurements that are at space-like separation in quantum field theory. The implication does not apply in reverse, however, so space-like separation of measurements is not equivalent to measurement compatibility.
The spanner in the works, a big one, is that measurement compatibility (and hence by implication space-like separation) does not imply no-correlation. There are correlations at space-like separation in quantum (field) theory, but one can prove (I realize here that I don't know precisely what additional assumptions are needed, but conventional QM frameworks are enough) that one can't use those correlations to send messages.
I have to point out that you shouldn't take too much for granted your parenthetical comment "(I assume I understand this one)". If you look at the other Answers here, you'll see that locality is far from simple. I particularly draw your attention to sb1's introduction of "influence" as part of his last paragraph's discussion, which I suggest is not simple at all.
It's important, IMO, to understand that this argument has been gradually changing over the last 50 years. It's not clear when or whether a novel argument will appear that makes it worthwhile to think in terms outside quantum (field) theory for practical purposes, but novel arguments are constantly emerging. The fact that Michael J. W. Hall (cited by Jim Graber above) has managed to publish his novel argument in Physics Review Letters is enormously impressive, particularly when one sees the robust tone he adopts, because PRL sets the bar very high indeed for foundations papers, but time will tell whether the argument can be used constructively in a quantum field theory context.
Finally, reading through my Answer, I realize that it does not directly address "realism" and "contextuality". That's because I associate mutual measurement compatibility of all observables directly with classical realism, all measurements commute, and the sometime presence of measurement incompatibility with contextuality. Measurements may have an "influence" (hee!) on some other measurements, and not on others. I might expand on this already overlong Answer later.
 Best wishes,
A: Realism refers to a philosophical position that says that certain attributes of the world of experience are independent of our observations. Let's take a physics example. In classical physics we used to say that a particle has a definite position and a definite momentum at a certain instant of time. These are represented by real numbers and they have those definite numbers independent of any observation. This seemed to be the only sane position one can take about the objective world. However the uncertainty principle of quantum mechanics tells us that a particle can not have both a well defined value of position and well defined value of momentum at the same time along the same direction independent of measurement. The more accurately one tries to measure one the less accurately one can have the knowledge of the other. Philosophically it means that position and its conjugate momentum can not have simultaneous reality. This realization had led the founding fathers of quantum theory to reformulate mechanics into a new theory called quantum mechanics. In QM a system is represented by a state vector in an abstract space. The length (norm) of this vector remain unchanged but with time its direction changes (for simplicity I am discussing Schrödinger picture). The various components of this state vector along the axes are various eigenstates with definite value of certain observables. Obviously the state vector is the linear combination of these eigenstates. Whenever a measurement is performed the state vector collapses to one of the eigenstates with certain probability determined by the Schrödinger's equation.
The so-called realists claim that the system was already in a definite state characterized by some additional hidden parameters before the measurement and since we are not aware of those hidden parameters we have an incomplete knowledge of the system. The random outcome reflects our incomplete knowledge of the system. There are number of hidden variable theories developed which reproduced the results of ordinary quantum mechanics.
Then surprisingly Bell discovered the famous Bell's inequality and showed that not all results are identical for both qm and local hidden variable theories. Experiment carried out and the verdict was clear. QM won. Nature supported QM. Therefore local hidden variable theories were ruled out. However there are nonlocal hidden variable theories which still survived like Bohmian mechanics. (I would also like to emphasize that MWI is an interpretation which is to some extent realist in spirit and it is by no means ruled out)
But what is locality?
Locality is the assumption that an object can be influenced only by its immediate surroundings by the events which took place in its immediate past. All classical and quantum field theories depends on this assumption in an essential way. Non locality implies that two events which are separated from each other by space-like separation can affect each other. Some people demand (imho) falsely that EPR type entanglement violates locality. In reality in never does. All one need to abandon is realism. Entanglement just shows that there exists quantum correlations between particles which were in past had some common origin. It also shows that if it were a classical world then the EPR entanglements effects were nonlocal. But we live in a quantum world and there is no non locality.
Therefore in a nutshell, locality is certainly not ruled out. Realism is ruled out to a large extent.
A: You may find these detailed lectures helpful. Particularly lecture 3 will answer your question. However, if you don't have the background you might want to go through all the three. http://qi10.ca/summerschool/speakers.html#FQM
A: I have nothing original to say about this. Unfortunately, Bell's result is the most misunderstood in all of physics. I leave you with quotes from Bell himself what the theorem means. 
It is remarkably difficult to get this point across, that determinism is not a presupposition of the analysis. (Bell 1987, p. 143)
Despite my insistence that the determinism was inferred rather than assumed, you might still suspect somehow that it is a preoccupation with determinism that creates the problem. Note well then that the following argument makes no mention whatever of determinism. … Finally you might suspect that the very notion of particle, and particle orbit … has somehow led us astray. … So the following argument will not mention particles, nor indeed fields, nor any other particular picture of what goes on at the microscopic level. Nor will it involve any use of the words ‘quantum mechanical system’, which can have an unfortunate effect on the discussion. The difficulty is not created by any such picture or any such terminology. It is created by the predictions about the correlations in the visible outputs of certain conceivable experimental set-ups. (Bell 1987, p. 150)
Let me summarize once again the logic that leads to the impasse. The EPRB correlations are such that the result of the experiment on one side immediately foretells that on the other, whenever the analyzers happen to be parallel. If we do not accept the intervention on one side as a causal influence on the other, we seem obliged to admit that the results on both sides are determined in advance anyway, independently of the intervention on the other side, by signals from the source and by the local magnet setting. But this has implications for non-parallel settings which conflict with those of quantum mechanics. So we cannot dismiss intervention on one side as a causal influence on the other. (Bell 1987, p. 149)
