Short answer--anyone that can get hired by a rink.
One of the things that all US Coaches must have in order to get "credentialed" (i.e. have permission to be rinkside with their skaters) is a membership in the Professional Skaters Association, or PSA.
Because no university, as far as I know, awards a bachelors degree in figure skating coaching (and you really have to wonder why not), PSA is the only game in town for a coach that wants to demonstrate training credentials in coaching.
For a profession that discourages the kind of parent coaching common in just about every other youth sport, the requirements for coaching figure skating are a scandal. Basically, there are none. Rinks are "accredited" not by the qualifications of their coaches, but by which type of curriculum they choose--ISI, USFS or none at all. In other words, there is no accreditation. Just membership fees. And anyone, anyone at all, that the rink will hire can teach figure skating. I know, because when I started coaching I had no teaching experience, was a very low level skater, and had never taken any test or coaching education.
Fortunately, I'm an ethical person, and I set out to get the credentials I felt I needed, and now have a Senior Rating in Group Instruction from the PSA and am also credentialed by the Ice Skating Institute as a gold level judge. For the PSA rating, this means I have passed three of the four levels of coach's credentialing in my chosen discipline of Group Instruction. To keep it active, I must attend an average of at least 9 1/3 hours of continuing education classes every year for as long as I want to keep the rating. You can get rated in numerous ice skating disciplines, including Group, Free Style, Moves in the Field, Choreography, Program Director, Figures, etc. I earn my hours by going to an annual 3 day seminar, by attending every ISI, USFS, and PSA seminar offered in my district, and through annual accreditation in Red Cross First Aid and CPR. (More about the ratings exams here.)
But it's not required. I never heard of a rink where you need to be rated to teach. Incredibly, not all rinks require First Aid or CPR training. You don't even need to be a member of US Figure Skating or the PSA to teach (although now you must be a member of both those organizations to sign your students up for tests and competitions). No rink I have ever worked for requires that coaches attend continuing education. I know of coaches who have never taken a single figure skating or coaching test nor skated in any competition. Rinks with formal programs to help Junior coaches learn the profession are few and far between; it's all watch and learn. Coaches that I have spoken with about this almost universally tell me that kids who can skate can coach. So, I can read, I guess that means I'm qualified to teach second grade! I can't imagine another profession where you jump from student to pro with no training in between.
I have a few problems with the PSA--I think that at the Master level the Group rating should be divided into developmental and freestyle; PSA seems to think that the higher your students are tested, the better you are. Which may be true for competitive skaters, but completely misses the point about group classes, where the vast vast majority of skaters are in the lower levels. For me, the Master rating in PSA, where they focus on questions about teaching the higher levels, is like asking someone who wants to teach grade school to pass a PhD level exam in particle physics. My specialty is beginners, and PSA should recognize the value of the skills needed for teaching beginning skaters by giving people like me our own rating exam.
It's a problem that PSA has no competing program; like I said, kids should be able to get a PE or Recreation degree with a focus on teaching figure skating or managing figure skating programs.
But PSA is the only game in town, and a PSA rating is a good indicator that your coach takes coaching seriously, understands that there are ethical guidelines, regards it as his or her profession, and does continuing education to stay on top of the sport.
I have asked Jimmie Santee, president of the PSA, to offer a comment for this post, but have not heard from him yet.