Financial resources for M.Sc. students in the U.S. universities

Most universities in the US provide financial aid to Ph.D. students. When the universities accept a student they inform him/her what are they offering (scholarship, stipend as TA, etc.). What is the situation with the M.Sc. students? Do they also get (sometimes) offerings about financial aid?

Maybe their offerings are based on resources that remain after some Ph.D. students decline their admittance. Is it possible for some M.Sc. students to get scholarships (from the university - not counting outside resources) or positions as TAs or RAs and thus they can pay their tuition and maintain a monthly stipend? Is the situation dependent on the student's nationality?

It is not an answer to a single question. I would appreciate it if someone explains the whole procedure that is followed for the M.Sc. students in the U.S. universities regarding the financial aid that they can get from the university.