I don't use them as rewards, more like party favors. Anyone who makes it through to the end of class gets to choose their sticker. The full-sheets of the same sticker makes that process much faster, otherwise you have to carry two or more sheets, so the early choosers don't disappoint the later skaters who wanted the snowlady with the purple hat.
The little "500 stickers!" reward books aren't a good buy - the stickers are too small to be enticing. Anything less than an inch square is sort of dismissed. The most favored stickers are shiny mylar ones that sparkle. I think they're usually labeled "hologram stickers."
Removable vs. permanent is a crapshoot. I've had removable stickers that just wouldn't stick to hands, gloves or helmets in the cold rink. They were everywhere - on the ice, on the floor, stuck to shoe bottoms and blades. Tip: never stick them on the kid's chest - hands and helmets are okay, but on the body is a no-no. (Although, one smart-aleck stuck it on his brother's butt during Mommy & Me. He thought he was hilarious and fell down laughing, literally!)
I buy seasonal stickers half-price after the season or holiday is over and use them during the following year. Christmas/Easter/Chanukah is dicey because you never know about religious issues. I try to stick with winter themes like snowmen or ice critters. Character stickers, like Disney or Pixar, are popular but if a kid doesn't know the character, they're not thrilled to get Thumper from Bambi, lol.
I've had a lot of kids turn down stickers. Some kids just don't like them.
I like the helmet stickers. We used to have the progress books for LTS with starskater stickers for each skill on the tests, but now we do the one-page sheets that's just printing. It's good for the higher levels, but the younger skaters prefer the badges - simple way to say "you did it!"
I keep stickers in my bag to use during public sessions. If a kid falls down or it's their first time skating, I ask the parents if I can give them a sticker. Distracts from boo-boos, as Slusher suggests, and encourages kids who might be a little skittish about the cold, hard ice.