Out of the three phrases, "Thank you very much" is by far the most popular. "Many thanks" and "Thank you very much" are both correct, but "Many thanks" is much more formal and would be used more in written form, at the end of a letter for example. I don't think it's grammatically incorrect to say "Thank you a lot" but no one says it that way. You could say "Thanks a lot" instead. Hope that helps! :-)

"Thank you very much" is the most formal of all. "Many thanks" sounds informal to me, too short. "Thanks a lot" is informal.

All are acceptable. "Thank you" is more formal than "thanks."