Here is the full list of all 17 Unicorns on Unicycles achievements.
Completed the tutorial
Defeat an enemy by knocking them to the laser barrier in the Anti-Gravity room
Defeat an enemy by bouncing them off the clock tower
Defeat an enemy by icicle impalement
Defeat an enemy by knocking them off the floating island
Defeat an enemy using the sea mines
Defeat an enemy using the wrestling ring ropes
Defeat an enemy by knocking them off the pirate ship deck
Loop around the space level 3 times
Bounce the same landmine 5 times
Survive the shaking sequence in the snow globe level
Launch off the top of the lava geyser
Survive 4 bell tolls on the clock tower
Clash horns 1000 times
Beat the game with 401 duels or more
Beat the game within 400 duels or less
Beat the game within 256 duels or less