In American usage, either is correct, and they mean the same thing- whatever is proper or just. For example, "the hosue is his by right(s)" (it is just and fair that the house is his), or "by right(s), you should be done by now!" (The expectation is that you should be done by now).

The correct expression is 'BY RIGHTS' and it means 'if things were fair'. E.g. It should be my house by rights as my parents wanted me to have it when they died, but they stole it from me!' and 'It should be my turn by rights, but she pushed in ahead of me'...'