Here are some of the
Not really a serious skater
A lot of kids start private lessons for help passing a level, or nailing a specific element. Then, when they have that element, they quit the lessons. Next time they need specific help, they get amnesia, completely forgetting the coach they worked with before. Coaches hate this. At least let the former coach know that you're going to work with someone else. Then when the inevitable "I thought Suzy was your kid?" question comes along, they're not caught blindsides.
"It wasn't a fit"
This is the number one reason given by coaches and parents when they can't give the real reason. The translation is (if coach says it), "that mother is a nightmare." If the parent says it "we realized that coach was not the fashionable one." Seriously, though, sometimes it really isn't a fit for numerous reasons-- schedule (see below), personality, cost, differing expectations. Of course, if it's one of these, just say so. Otherwise people will assume one of the first reasons. (Better yet, if asked, just say "we decided to make a change." If pressed, continue to say this, for which the translation is "it's none of your business.")
UPDATE: There are a lot of replies in the comments suggesting that ALL reasons fit under this excuse, including gross unprofessionalism including habitual tardiness, inappropriate attitude ("being mean"), emotional abuse ("you're stupid, you're fat), and financial shenanigans (short lessons, full price). Folks, when coaches act like this it is everyone's business. There is absolutely no reason to hide behind social lies when there is a legitimate consumer reason for the split. If you were habitually overcharged at a store, or yelled at by the clerks, would you tell people you don't go there because "it wasn't a fit?"
This is a tough one. In a market like mine, schedule is a non-reason. There is so much ice that you will be able to find the time. Plus, I've often had parents tell me the schedule won't work, just to see them with the new coach on the exact ice I offered. In smaller markets, schedule (which includes necessary travel time) does sometimes necessitate a change in coaches. However, you cannot blame the schedule if your idea of scheduling is "we can have a lesson during this single 30-minute window and are not flexible on this." If that is the case, you're not looking for a coach, you're looking for a babysitter.
For competitive students, all other things being equal, it makes no sense to change a coach who has been successful with your skater (as measured not by wins, but by accomplishment, skill increase, and personal best scores). If you're skater's doing well, but you switch anyway, then you're a "it wasn't a fit" parent.
Number one reason people drop in and out. See "not a serious skater" about which coach to go back to.
If you're making your ambitious skater switch coaches every 6 months or more, you are destroying your skater's career. For one thing, strong competitive coaching relationships take years to build. For another, different coaches have different, usually equally acceptable techniques and practice protocols. You will lose a season every time you switch. Further, if you're showing up at the non-quals in late summer with Coach Success, then turning up at Regionals with Coach Whosit, but then at Nationals with Coach Fashionable, the judges are just going to roll their eyes and move on, unless you're really blowing them out of the water. Which if you switch coaches like this, you won't.
Finally, to zero in on the serious side of this issue:
If this is your reason for switching coaches, for god's sake tell the skating director, or if you don't trust the skating director, then tell your doctor, and be prepared to back it up with documentation (names, dates, places). Involve the skater in this decision. Abuse includes inappropriate touching, questionable language, insistence on excessive dieting, and emotional abuse.
The downside of constantly switching coaches is that people stop taking the skater seriously. One of the most common sights at the rink is the talented 17 or 18 year old who has never made it out of the preliminary rounds at Regionals but has decided to put off college to give it one more shot, with her 8th coach. We all shake our heads and blame the parents.
I'm not saying never switch coaches. My daughter had 4 coaches in her 11 years of semi-competitive skating. Her first coach moved to Florida. Her second coach quit to run her family's business. She developed a serious personal dislike for the third one, for reasons which did not become clear until years later. She still works with her last coach (going on 10 years).
Do the math. Are you a coach hopper? What's your excuse?