I've recommended auditioning coaches, and a reader asks "can I really do that?"
And, yes, of course you can. It's your money, time, and child (or self). Depending on where your skater is in her "career" there are different ways of going about this.
For very young children and beginners, the best thing to do is simply hire a coach that your child (or whoever the skater is) gets along with well. Don't worry too much about credentials-- if they've been hired by a rink, they have the credentials they need to teach a 5-year-old, pretty much. Your first coach does not need to be the one that gets you to the Olympics, he needs to be one that inspires a love of skating. You've already "auditioned" this coach in class. Auditioning several coaches at this level is just going to look silly, and furthermore is going to give all the coaches in the rink the idea that you're a bad risk.
In fact, class is the place to start the audition. If you move to a new rink, and don't come armed with a referral from your old rink or coach, just take class with as many different coaches as you can and see who clicks. Then ask a couple of the coaches if you can try a lesson with them, making sure that every coach KNOWS you are trying out different coaches. I wouldn't do this with more than 2 or 3 of them. Then choose one; the other ones will grumble and gripe for a while and then forget about it. As a species, figure skating coaches have very thick skin and a marvelous ability to rationalize not getting students.
I'm pretty sure I've been "stealth" auditioned--I'll do a couple of lessons with a kid and then come in and see them working with someone else. Fine, whatever, but do me the courtesy of telling me this is going on.
You don't have to hire the first coach that approaches you. If you're at a rink with those "lesson request forms" remember that only one coach ever sees it-- the first one that grabs it off the board, or the first one on the skating director's "next up" list. Tell them, sure, let's try out a lesson and see how it goes, but that phone call does not commit you to the lesson. By the way, if you decide not to go with this coach and the sheet goes back up on the wall, rethink it, because that's an ethical and generous person. (If you turn in a sheet like that and don't get ANY calls, there's a red flag on there somewhere-- more in another post.)
For competitive and higher level skaters I really believe that auditioning a coach is imperative. You need to know that you like this person's teaching style, that you are clear on the financial and time commitment, and that you're going to get what you need personally and professionally. At this level, it's not only taking a couple of trial lessons, but also observing the coach with her other students. At nationally competitive levels, this works both ways--the coach may be auditioning you as well, and will agree to take you on or not.
Higher level and seriously competitive students should also not limit themselves to the staff at a single rink or program. If you're that serious, you need to know ALL the options in your area, and not just the ones at the facility most convenient to your home. If convenience is your number one criterion (in fact if it's even on the list), you may want to rethink that whole "serious competitor" thing.
If you are working with a regular coach already, and are thinking about a change, do NOT audition other coaches without letting your current coach AND the prospective coaches know. And, once you tell the current coach about this, then the relationship is effectively over, because the coach knows you're unhappy. Try to repair the current relationship first.
More on choosing a coach:
Young Coach or Old Coach
How to choose a coach: Part 1 Part 2 Part 3