While some students may be eager to work with a famous full professor, others might fear that the busiest advisors would have the least time for their students. Finally, be aware that procedures for matching students and advisors may vary by program or department. To select the best advisor, you might meet with all the faculty members of your program and talk with other gradute students in the program about the qualities of the faculty members eligible to direct dissertations. Be cautious about making assumptions, and ask questions covering a range of topics. Some questions should best be discussed with the faculty member in question, others might better be asked of advanced graduate students.
Some items pertain more to specific disciplines than to others.
Remember, too, that faculty members will have a number of questions to ask about you.