For bipartite scenarios we have CHSH inequality which is a (2,2,2) inequality,  implying 2 parties,  2 measurement settings with 2 outcomes.  It is the only (2,2,2) bell inequality.  For (2,2,d),  d outcomes,  there is the CGLMP inequality.  For three parties,  I think there is the I_{3,3,2} inequality (it goes by this name only).  Well,  these are the popular ones.  For references to more inequalities I would refer ypu to the review article by Brunner et.  Al..  In section IX or X of their paper they give a review of multipartite Bell inequalities.

I did not mention before,  but Bell inequalities are tests for non-locality.  Are they the only tests?  According to my knowledge they are the only one.   

We work a lot on this. We had a preprint on efficient nonlocality detection recently, see arXiv:1612.08551. Bell inequalities are convenient because they define polytopes, but they are not the only way to go for detecting nonlocality. Another way is to study the ground-state energy of a many-body system; see, for instance, arXiv:1607.06090.

https://arxiv.org/abs/1612.08551

https://arxiv.org/abs/1607.06090

Thanks Jaskaran  and Prof. Wittek. I'll work through the arXiv papers. Regards Aman!