Let's say I am having a conversation with an university student and want to ask about his or her major. In that scenario, which one should I use, "what are you studying" or "what do you study"?
Both are grammatically correct. However, "What are you studying?" is referring to a current, ongoing event, and therefore I would use it to a person who is currently studying, and expect not the major but the particular class, which might not even be in the major.
"What do you study?" is ambiguous. On first hearing, I would not be sure what was being asked. It could mean "what subjects do you have to study for (instead of breezing through)?" and I'm not sure I would think it asked for my major.
The best question would probably be, "What are you majoring in?"