All accounts (of the same gender) get the same moves at the same levels, so there's no chance a player the same level as you or lower can get a move you don't have. Guys and Gals have a slightly different list of moves, so you can't compare those two lists. Both guys and gals get mostly the same moves, and they get the same number of moves.
To view and change the quick selection button Dance Moves at the bottom of your screen, you need to get into the "Bust a move" section of the game.
Click the arrow in the bottom-right corner of the screen. You'll see some icons appear. Clicking middle "Bust a move" icon (looks like a person) will replace your chat log with a display of all your Dance Moves. From there you can drag the moves you like down to the quick select section of your toolbar so your five favorite moves are available when you just gotta shake that thang.
If you want your Chat History display back (and you probably do), just click the Chat Balloon icon right below the "Bust a move" icon.