Develop an email conversation with them. Talk to them if they express interest in mentoring you.
Meet with them if at all possible before you apply or accept your acceptance to their grad program. Talk to other students about their impressions. You're assessing the profs and departments at least as much as they are, you. Look critically at their research publication record. Are the articles, no matter on what topic or in what quantity, diverse and insightful. Or are they essentially reproductions of the same method and topic? Quantity or quality? This will give you an indication of how open and critically minded the prof is and thus what they may offer you in terms of intellectual mentorship.
Are they good enough to work with you? Do they treat you like an adult? Do they sound like how a used car salesmen treats customers or is there a sincere an engaging professionalism in their dialogue with you?
2. When looking at graduate programs, try to visit the department and gauge the mood or atmosphere in the place. Does it feel stressed or hostile? Or is it friendly and welcoming. Walk away, or risk several years of misery.
3. Funding. Make sure your department and supervisor are unambiguous about the if and how your financials. You should be left in no doubt about the amount, duration and classification of your funding. If you can come with your own, even better. If you sense they are being opaque or unclear on any aspect of it and cannot get clarification, walk away. This is a dishonest department.
4. If they are clear on funding, double check with the university or graduate student association to see if what they tell you is in line with what the departmetn tells you. The gradstudent association may even have a Collective Agreement with the university over how you're paid. If there is a discrepancy between the your program and the rules, walk away. This is a place that exploits its students and subverts the rules.
5. KNOW the funding rules in advance so you can recognise early enough on when they're trying to exploit you. You may not realise it, but you are likely a member of a union (this often surprises gradstudents) and have clearly spelled out rights regarding your work life.
6. Have a look at the typical completion time of recent degrees in your stream. If they promise you a two year masters or 4 or 5 year PhD, but all the gradstudents you meet have been there much longer than the stated norm, walk away. This is a department that has lost the plot.
7. Just because the supervisor has a really big name in their field doesn't mean that they are a good supervisor. Sometimes their ego is even bigger than their CV and they have come to believe they are the alpha and omega of their field. I remember being at a conference where the BIG NAME keynote spent literally half his talk describing the
8. Walk away from and/or call out bullies, liars, thieves, creeps, misogynists, misandrists, misanthropes (well some of them, others are lovely), micromanagers, or any peer, colleague or supervisor who makes you feel uncomfortable in the bad way. Their status or education does not mean they are of better character. And universities hire on CV and grant records, not personality.
9. If you feel your supervisor is coopting the thesis topic they agreed to mentor, to fit into their own paradigm or views, challenge it. You are turning into their research assistant, not their student. If they can't take a respectful intellectual argument from their student, walk away.
10. If your supervisor doesn't understand or is uninterested in your work, they are not mentoring. In some extreme cases, this may represent academic incompetence. Find an alternate arrangement up to and including changing supervisors.
11. Use your committee. They can also critically assess your work if you suspect the number 10.
12. Trust your sixth sense about situations. If you begin to have doubts about the capacity of your supervisor to mentor you effectively, listen to this and suss out a contingency plan should the worst bear out. You won't likely lose your degree but you may have to make some courageously proactive and decisive manoeuvres. Changing supervisors might be an ordeal but there are mechanisms in place to do this.
13. Start documenting things if your sixth sense starts to tingle. Keep a paper trail of emails, comments, etc. Write summaries of your interactions with the party in question.
14. Understand that if you do decide to challenge an individual or organisation at the university, the institution will likely close ranks and resist. Most professors have a monetary value, graduate students are a dime a dozen. In a neoliberal university, capital holds the immediate power. If they think they can crush you or slough you off, they will try. Assume that unless that prof is universally hated by their colleagues, and/or doesn't bring in much grant money or publications, you might be in for a struggle. You MUST be prepared to follow through on your course of action.
15. Worst case, you can walk away from your degree. Life might actually get better.
16. In sum, find in as much as possible the best mix of people, place, and institutional atmosphere for your tolerance levels. Understand that higher education is often extremely arduous and filled with stresses, but the difficulty should mostly pertain to the intellectual realm and not the interpersonal, financial, or administrative.
You do not want to commit several years of your life to a place where you'll be miserable and without joy. You'll gain little more than bitterness from the experience.