Putting aside the issue of obviously automated inquiries, what should I do with personalized emails I receive from students asking to be my PhD student? Assume that I know the student is not going to become my PhD student any time soon, possibly because there are no vacancies currently. In particular, I am interested in departments where PhD admissions are not centralized.
If your department or institution has an established centralized procedure for admitting graduate students, I would prepare a template that says:
All interested students must apply through (insert method here). I do not respond to direct unsolicited requests for PhD positions.