I think the most important is: * Are you happy with your PhD in general? It covers a lot of issues, but usually you don't want to do a PhD in a place where students are unhappy and frustrated. The good thing is that is students feel really bad, they rarely hide it. The next things are related to: * General contacts on the line student-advisor (How much contact and support can you expect, both for research and administrative stuff?). * Funding (Is it a problem or a non-issue, e.g. for attending a conference?). * Research (What the actual research looks like? What is the toughest part, biggest emphasis on, the most time-consuming part, etc?) * How much time does it _typically_ take to finish PhD? Does it happen often that someone drops out? Other questions will depend of things _you_ consider important. You may expect a lot of autonomy, or a lot of guidance. You may expect a very ambitious programme, or a PhD-life balance. You may like to teach, or you may like to keep it at minimum. You may be eager to travel a lot, or prefer to stay mostly at one place. You may be more into particular branch of science or methodology, or into other...