Here is the full list of all 14 Beardy the Digger achievements.
Collect 16 diamonds
Collect 32 diamonds
Collect 64 diamonds
Complete summer levels with 50 or more diamonds collected
Complete winter levels with 50 or more diamonds collected
Complete a level while being affected by any potion
Use a catapult
Use a catapult while being affected by a potion of poison
Die by lava 4 times
Let moving cubes crush Beardy
Complete a level without using a key
Complete 5 levels with only 1 diamond collected in a row
Keep balancing on ice for 30 seconds
Start the mini-game from the main menu