Here is the full list of all 39 Chess Ultra trophies - there are 26 bronze, 7 silver, 5 gold and 1 platinum.
Have two pawns of the same colour on the same file
Castle kingside
Pin a piece against your opponent's King
Play a game using the Fischer timer
Deliver checkmate where the King is blocked on both sides by his own Rooks
Put an opponent in check
Win a game
Beat an opponent with an Elo rating higher than yours
Make a capture or exchange
Unlock all other Trophies
Stack two Rooks one behind another in the same file with the Queen at the rear
Deliver checkmate by a Rook or Queen along a back rank, with the King's escape blocked by friendly pieces
Castle queenside
Advance a pawn to the eighth rank
Promote a pawn to a Rook, Bishop, or Knight instead of a Queen
Perform an en passant capture
Deliver a check by two pieces at the same time
Complete all the tutorials
Capture 100 pieces
Win a game in three moves
Win a game in four moves
Place a White Knight on e6
Develop a White King Bishop to b5
Win a tournament
Complete 50% of the chess challenges
Create a random online game
Issue an online challenge to a friend
Sacrifice the Queen then go on to win the game
Using a Knight simultaneously attack the enemy King and Queen
Beat an opponent with an Elo rating more than 75 higher than yours
Win 5 tournaments
Maneuver a pair of Rooks onto your opponent’s second rank
Draw a game by stalemate
Deliver checkmate by two crisscrossing Bishops, with the King's escape blocked by friendly pieces
Win a 32 player tournament
Beat an opponent with an Elo rating more than 150 higher than yours
Beat the computer on Grandmaster difficulty
Complete all the chess challenges
Win 20 tournaments