Costumes replace the old outfit system from the Flash version of the game.
Unlike outfits, costumes are only cosmetic; they don't affect digging at all, only what Diggy looks like.
Currently, several basic costumes are available for all users (see below). Current Diggy is the Diggy introduced with WebGL; Retro Diggy is from the Flash version. There is no costume (yet?) for the old Mobile Diggy.
Event costumes are the first item unlocked in the premium row of event passes; once claimed, they are available permanently.
To switch costumes, go to the Costume tab in your inventory.
These costumes unlock after the first quest in each region.
These costumes have been in the shop for a limited period