While playing a puzzle, coins can be used to purchase Boost Cards by tapping their icon at the bottom of the screen when you have none of that card in reserve (the icon will be marked with a '+' symbol.)

Coins can also be spent to bypass Quests and reveal Trophies, by accessing the 'Quests' or 'Trophies' screens from the Progress menu (accessed by tapping the purple icon in the status area).