By "classical" I mean not quantum, so classical mechanics, electrodynamics and relativity. Perhaps a different label is more appropriate.
First of all, on Wikipedia it is claimed that "Local realism is a feature of classical mechanics, and of classical electrodynamics", so I assume also of SR. However, I can't quite square that with my understanding of Newtonian mechanics. As far as I know, Newton postulated five things: Absolute time, absolute space and his three laws of motion (very roughly put). Add the gravitational law, and you get accelerations that are caused by forces acting instantaneously at any distance. Is this wrong? If not, then how can Newtonian mechanics be local? I might also be misunderstanding the principle of locality or local realism.
Secondly, in classical electrodynamics. For instance, Griffiths claims that "electromagnetic 'news' travels at the speed of light" (Introduction to Electrodynamics, 10.2.1). First of all, this is consistent with locality, since there is a finite speed at which the information is transferred. But it seems a stronger claim than that, since the news are claimed to travel at a particular speed.
I am not clear on the exact postulates of classical electrodynamics (apart from Maxwell's equations and the Lorentz force law), so I have some trouble seeing exactly what the justification for this claim is. Griffiths has previously shown that electromagnetic waves propagate with the speed of light (at least after taking into account the experimental discovery that $\mu_0 \epsilon_0 = 1/c^2$), and I can sort of see how that might be relevant when discussing retarded potentials. But lacking a rigorous definition of "news" or "information", it is not clear to me if this is enough to conclude that information also propagates at this speed.
Griffiths even considers advanced potentials, but claims that they are non-physical since they violate the principle of causality. But as usual he is very vague and calls it the "most sacred tenet" of physics. Do we then, in classical electrodynamics, postulate time-asymmetry? Stepping outside of this theory for a moment, I was under the impression that the asymmetry of time (at least up to a CP transformation, which ) is a purely macroscopic phenomenon due to the second law of thermodynamics, which itself results from the boundary conditions of the universe, i.e. a very low entropy at the Big Bang. It makes sense then that, inside a supposedly self-contained theory such as classical ED, we would take it as a postulate, since we cannot appeal to theoretical results outside of the theory. I find this satisfactory enough, but I would like it to be stated explicitly. (Also, in the light of the above, Laplace's ideas of conservation of information, and in particular Hume's critique of causation, I find it a bit unsatisfactory to brush all discussion aside and simply take the principle as self-evident. There might be something I'm missing.)
Thirdly, special relativity. I find relativistic dynamics very confusing, but to my knowledge relativistic kinematics only contain Einstein's two postulates. And the dynamical theory of SR doesn't seem to postulate much else, only define relativistic generalisations of classical observables such as energy and momentum which reduce to their classical counterparts in the non-relativistic limit. It takes on further experimental results, such as the conservation of four-momentum, but I don't know that that is relevant. I might be wrong, of course. Thus it is again unclear to me exactly what the status of locality is in SR, and why we should assume that the speed of light puts an upper limit on the speed at which we can transfer information.
Finally, since I assume that this is a topic much too large to be explicated here, any literature recommendations would be appreciated as well.