Which of the known fundamental laws (axioms / principles / equations / inequalities) of physics are absolutely non-negotiable (cannot be violated without causing major inconsistencies or singularities) and which are negotiable (can be slightly relaxed without blowing up the whole physics) ? For example, violating the equivalence principle within the experimental bounds seems to be fine but violating causality, no matter how slightly, would lead to absurdities.
I think this is an important question to think about especially when we are faced with situations where the laws of physics seem to be incompatible with one another, e.g. when attempting to construct quantum gravity, and we are forced to modify some of them. We need to know which "screws" can be loosened before can even begin molding the relevant parts of physics with the intention of making everything consistent.
The best type of answer that I am looking for is one that lists useful references studying this subject thoroughly. But, my impression is that the literature seems to be lacking studies on the subject. I could find many studies on violations of Lorentz invariance but not so much on others. So, I would also be happy if you could list some fundamental laws of physics and argue what would happen if they are violated slightly. Preferably, the discussions should only be within the established frameworks of physics: QM, GR, QFT, etc. I do not know enough to appreciate arguments based on, say, String theory.