Do grad students in the USA typically have more or less student debt than undergrads? A student advocate at my institution recently claimed that grad students have more debt than undergrads. This surprised me, because I would expect that typical grad students are the kind of student that would be able to gather more financial aid (scholarships, awards, fellowships, etc.) during their time as an undergrad than typical undergrads. Furthermore, most PhD students (and many masters students) have their tuition waived while working as a TA/RA, so they accrue little/no additional debt. Of course, this likely varies with field, too: students in STEM are more likely to have institutional funding while law or medical students often have to pay their own way. 

While I don't know if the statistic I quoted was comparing undergrads who haven't yet accrued their full measure of debt to grads who have all of their undergrad debt, it begs an interesting question:

Do undergraduate students typically have more student debt than graduate students have, and (broadly) how does this vary with field of study?

The comparison ought to be made after each group graduates from their respective program, since comparing a 1st year undergraduate's debt to a 1st year PhD student's debt clearly would not be helpful.
I think you are underestimating the number of graduate students who do not get any form of financial support. Like you said, most students in STEM or engineering PhD programs full receive full support. PhD students in humanities/social sciences departments are less funded compared to STEM students, but at least are given tuition remissions. 

On the other hand, majority of MA students in any discipline generally receives no support. Then there is MBA students, law students, doctoral residents etc. They are the ones who make up a big portion of all graduate students and most of them have to get student loan to complete their graduate programs. 

For example, a typical law school graduate would end up with 60k-100k+ loans at the time they pass the bar and get a job.