Assuming you get to choose.... In some programs, advisors are assigned by the faculty, and the student has little or no say. The official reason for such policies is to ensure that workload is equitably distributed across faculty members; otherwise, some faculty tend to end up with dozens of graduate students while others have none. Of course, from the students' perspective, this reflects the fact that some advisors are wonderful mentors, while others would not be voluntarily chosen by any knowledgeable student.
In other programs, you choose your advisor before you apply—you make a connection with the potential advisor months in advance of the application deadline, and that advisor advocates for your application on the understanding that if you are accepted, you will be that advisor's student. (The corollary is that if you are not accepted into a particular program, it may not reflect on the quality of your application; it may be that the admissions committee simply decided that Professor X already had more than their fair share of students, and to admit Professor Y's student this year instead....)
In most programs, you apply and are accepted, and have a choice to make. Choosing an appropriate supervisor (and the other members of your committee) is a key decision in successfully completing your thesis, dissertation, or final project.
When you are considering potential supervisors always talk to several professors before asking one to become your supervisor. Even if you initially chose campus X because it had a really strong scholar with whom you wanted to work, you may find it worthwhile to consider what your alternatives may be. Although it may be tempting to simply approach the first professor you encounter in a course or someone else identifies might be good, you need to carefully evaluate prospective supervisors and committee members on the basis of objective criteria. Even if a professor approaches you (usually a good sign!) before you have had a chance to fully research your options, simply say that it is too early to make a decision. Put at least as much time into researching and choosing an appropriate supervisor as you would into buying a car.
[Be sure, however, to clearly distinguish between these preliminary, exploratory interviews and the actual request to become one's supervisor. You do not want (as happened to one of my graduate students) to talk to five professors and later discover all five are under the impression you asked them to be your supervisor! Start the conversation by clearly stating that you are not ready to choose yet, that you are only exploring a number of options, and that you still have several others to speak to yet. Briefly outline your idea and ask who they would see as the best choice. Pose questions related to the criteria below. Thank them for their time and opinion. Tell them you will let them know by such and such a date if you intend to proceed along the lines discussed.]
Criteria for Choosing a Supervisor/Committee Member
The professor's area of expertise could contribute usefully to the project.
A knowledgeable professor can save you thousands of hours by quickly identifying key concepts, resources, or personnel; can stop you from going down blind alleys; and can provide the quality control that ensures that you will be proud of the finished product.
The professor is available.
Is s/he going to be on campus during the whole period of your study or might a study leave, summer vacation, or other commitments take them away just when you will need them most? Even if physically present, will they be able to devote the time required to help you? How many other graduate student's has this professor taken on for the same period? How many committees? Their own writing, research, consulting, or performance activities? Are they prepared to commit to a regular meeting time with you (e.g., for 90 minutes the second Tuesday of every month?) Be prepared to discover that, with increasing graduate enrolments and instructor workloads, some professors will not be available to you because they have already committed to other students.
The professor's style, personality and standards are a close fit with, or complementary to, one's own.
Although it is quite possible to work professionally with someone with whom you have little in common, finding someone with whom you work well does make completing the thesis or project considerably easier and more pleasant. Compatibility may be difficult to ascertain before hand, but remain alert to these criteria in your initial sounding out of prospects. Supplement your own initial impressions with feedback from other graduates, instructors, and colleagues. (Remember, though, that one person's flaw is another's virtue—an advisor who appears to one student as an anal retentive with impossible standards is another's ideal guarantor of a quality product, and vice versa.) Remember also that compatibility cuts both ways: you may have to convince potential supervisors and committee members that your style, personality, and standards are compatible with theirs . . . .
The professor is prepared to work with the others you have decided you want for your committee.
This inquiry requires a certain amount of tact, but it is crucial to avoid placing feuding academics on your committee, lest you get caught in the crossfire as each insists the other's approach/advice is all wrong. If you are happy with your choice of advisor, you may want to accept their advice on who else to seek for the committee. If, however, you definitely want a certain individual for the particular expertise or personal contacts etc., they can bring to your project, then screen your other recruits for their ability to work with this person. (Not, Robert hastens to add, that there were any such feuds on my campus. *Cough* Cough*)
The professor has a proven track record.
How many theses or projects has this individual supervised? Did students complete their studies in a timely fashion? Were this individual's alumni satisfied with their supervision? It might also be useful to review the professor's own publishing record.
Useful Sources of Information
The following sources of information may prove useful in evaluating potential committee members in terms of the particular needs presented by your specific thesis, project, or independent study.
"Faculty Research Interests" Documents
Most faculties identify their faculty's areas of expertise, either in the calendar or in a separate document available to students on-line or through the Dean's or Graduate Program office.
CVs are generally an excellent source of information. In addition to listing the professor's degrees, work history, and courses taught (all good indicators of expertise and interests), they often include a list of thesis and projects supervised. The topics previously supervised again provide some indication of the instructor's interests and expertise. The length of the list also indicates how much experience the instructor has had supervising theses, projects, and independent studies. Finally, the list of authors may help you to identify alumni who might be queried about their supervision experiences with this professor. (If an instructor does not make their CV available, you may wish to inquire into their reasons and to adopt a "buyer beware" attitude.)
Published Articles, Fiction, Poetry, Reviews, Webpages or Performances
An obvious and important step in selecting and working with a supervisor or committee member is to review at least a sample of their own published work. Do you find their ideas interesting, their research credible and worthwhile, their writing accessible? (After all, these are the qualities you are hoping they can help you develop as they guide your study.) Is their research, theorizing, methodology, or vision relevant to your study? Is their style, approach, and stance sufficiently compatible that you wish to work with them, or might you find yourselves constantly at odds over how your study should proceed? Would you like to be known as this person's student or protégé?
Have you taken any courses from this person? Sat on any committees with them? Direct personal experience in another context is often a useful supplement to a review of a professor's published work, particularly assessing compatibility in terms of personality, work styles, etc. One should be cautious, however, of signing on with the first interesting instructor one encounters before exploring all options
Other Instructors, Colleagues, and Alumni
Approachable course instructors are often a useful source for identifying potential committee members appropriate to one's particular topic. Any such recommendation must, however, be placed in context of the speaker's biases.