Yugioh, in the USA, most adjunct professors work on semester-long or year-long contracts. Some are not even able to say right now, in January, whether they'll be employed in August, much less whether they can supervise a PhD student. You should not apply to do a PhD supervised by an adjunct in the US.
Emeritus professors have retired and generally do not take PhD students since they are no longer full-time faculty.
Adjunct Professors in other countries: Fulltime Professor in one instituton (has full-research group there) who are invited to work part-time in other institution.
In these countries, you can have adjunct professors as your advisor with conditional requirement that you have another full-time professor in your institution as co-advisor.
Adjunct Professors in USA: Part-time worker scrambling for work
Adjuncts are usually part-time, supplementary professors to the program, and generally teach at several universities in proximity to one another in order to make ends meet.
Only graduate faculty are allowed to advise graduate students. To be graduate faculty, you must be a full-time professor with an appropriate PhD and in good standing. Professors who are in phase-out retirement standing cannot serve as advisors, nor can "junior professors" (MA, ABD holders), and certainly not adjunct faculty members, who may or may not be rehired from one semester to the next dependent upon financial and course-related exigencies.
The website for a university, or the graduate catalog of the university or department, should have a list of faculty members you can approach as graduate advisors.
You CAN have an adjunct professor on your thesis or dissertation committee; in fact, for the dissertation, you must have at least one outside scholar as a member of your committee. But only if the professor has a PhD. MAs are usually not permitted to serve as graduate faculty or on graduate/thesis committees except under exceptional circumstances and with department approval ahead of time (that would be the blind leading the blind, yes? Or like being reviewed by a coworker instead of your supervisor). But they absolutely can't be thesis or dissertation advisors.
FYI, because it doesn't hurt to know, the order is:
Assistant Professor (junior faculty/ pre-tenure - can advise undergrads and sit on committees for graduate faculty; with appropriate degree, can serve as a graduate advisor (PhD only))
Associate Professor (usually awarded with tenure, must have a book)
Professor (highest full-time position in a department; often an endowed chair, but not always)
Emeritus (retired, but can still teach an occasional course and/or sit on a committee)