Here is the full list of all 36 Battle vs. Chess trophies - there are 20 bronze, 11 silver, 4 gold and 1 platinum.
Earn every trophy
Win a single game in every game mode (not Head to head)
Reach an online rating of 1200
Win 20 games online (not Head to head)
Win a game while possessing 2 pieces less than your opponent (Classic Game, not Head to head)
Win online game against the player with a higher rating
Win a game in less than 30 moves (Classic Game, not Head to head)
Play a game to Stalemate (Classic Game, not Head to head)
Earn a Gold Medal on every mission
Move an opponent into Checkmate using every piece (not Head to head)
Earn at least a Silver Medal on every mission
Complete all challenges, puzzles and missions (not Head to head)
Complete 20 levels of the 'Treasure Hunt' challenge
Complete all puzzles
Play 100 games (not Head to head)
Win a game in less than 5 minutes (Classic Game, not Head to head)
Promote your Pawns to every possible major piece (not Head to head)
Place your opponent in Check by Castling (Classic Game, not Head to head)
Capture a Pawn 'En passant' (Classic Game, not Head to head)
Move your King to the opposite side of the board (Classic Game, not Head to head)
Earn at least a Bronze Medal on every mission
Complete 15 Chaos missions
Complete 10 Chaos missions
Complete 5 Chaos missions
Promote 3 Pawns in a single game (Classic Game, not Head to head)
Place your opponent in Check using 5 different pieces in a single game (Classic Game, not Head to head)
Place your opponent in Check 10 times in a single game (Classic Game, not Head to head)
Capture 5 pieces with the same piece during a single game (Classic Game, not Head to head)
Capture all enemy pieces and then checkmate the king (Classic Game, not Head to head)
Win a Classic Game by Checkmate (not Head to head)
Beat a Level 6 AI opponent in a Classic Game
Complete 15 Order missions
Play over 1000 turns (not Head to head)
Complete 10 Order missions
Capture 100 pieces (not Head to head)
Complete 5 Order missions