I'm a supervisor not student these days. I had things go rather wrong as a student, quit my project and moved to a new team and restarted everything. I knew about my proposed supervisor's field reputation (excellent - I still think this even after things went wrong) - I knew nothing about what they were like as a supervisor (they sucked).
This also should be more a conversation over coffee - you don't want them to feel like they are being grilled or are under evaluation from you. Remember, if you accept an offer they could be your colleagues so you are selling yourself to them too. Keep things positive on your end.
How would you describe their supervision style? Different styles suit different students, know about yourself as best you can. I know of advisors that have contracts and forms that students fill in after each meeting summarising their understanding of the key points and next steps that advisor then responds to and they both sign -> student speaking of this loved it, I would have run screaming.
How would you describe the group dynamics?
You usually aren't just working with the supervisor but the group. Even if your project does not involve the group - these are your colleagues. How do they tend to function? Is it everyone on their own? Collaborative? Collegiate but not collaborative?
Did you have much / a lot of freedom to direct your own project?
What sort of involvement have you had in funding proposals (grants / commercial etc.)
This can be good and bad - too much involvement and you are taken away from your core business. But some involvement can really help develop skills useful for industry and academia, and can help land that next position.
Does your supervisor include you in opportunities. Will their "big name" actually get you anything? Remember "big names" are often super busy, so there needs to be a trade off (big names do help just by reputation, but it isn't everything).
How involved are you in the larger group / School / Institute / University?. Just useful to know how they tend to work, and again knowing what sort of person you like to be. If you just want to focus on your project and nothing else, being expected to engage widely will feel like a drain. For others it is great career development opportunities (and helps them stay sane while studying).
What are your career goals - how well have they been supported through your studies? Have they changed through your project
Helpful to understand their perspective. Also good to know if the supervisor is super helpful if you want to become a copy of them, and not helpful if you have different goals.
What's a typical work week look like for students (hours, office vs lab vs field, wfh vs in office, group meetings, paid work etc.)
This is important information about work style and allows you to evaluate whether you will have a nice work environment (based on what you value) and good work-life balance (based on what you value). If your supervisor says I expect people to work 9-5M-F (or set your own hours), but the typical is 8-8M-Su then that's really important information. Even if your supervisor is happy with you doing a "standard" day, will you feel comfortable keeping that balance when everyone else isn't? How much "other" stuff are students typically doing and how will that affect your plans?
With paid work (teaching, tutoring, research assistant stuff that is not on your thesis) - how well does the number of hours you are contracted to do match to what you are expected / end up doing?
If you pick up 10 hours teaching a week will you typically spend 8 hours on it? or 20? Important to be able to make an informed decision about what you want and can afford. You might earn double what you'd get at Maccas so might be fine that 10 on paper = 20 in practice cause you still earn as much and are happy with the work, or you might resent what really is wage theft and an illegal, dishonest and exploitative practice.
What's the administration of your studies like eg reports, presentations, hurdles etc.. No external oversight is a problem - you do need some outside your supervisor review that your project is at an appropriate standard, is practical and realistic with resources (including time), and you aren't being hung out to dry. You need some way to raise issues and have people whose job it is to deal with them if they happen (and they are not uncommon - even with good supervisors). But, too much administration and it becomes a burden. Understanding what is required is just nice additional information. It probably won't be the same for you - universities love to change up their admin frequently in my experience.
Question about the project and funding others have proposed are also really important, I've focussed on the "other" stuff that often gets missed.
