Here is the full list of all 49 Dragon Ball Z: Battle of Z trophies - there are 34 bronze, 12 silver, 2 gold and 1 platinum.
Obtain all Trophies
Collect all cards
Attain 50 hours of mission play time
Without using the same characters, make a team of 4 from the various Gokus and win in Battle Mode
Without using the same characters, make a team of 4 Androids and win in Battle Mode
Without using the same characters, make a team of 4 Saiyans and win in Battle Mode
Without using the same characters, make a team of 4 from Frieza's army and win in Battle Mode
Without using the same characters, make a team of 4 Z Fighters and win in Battle Mode
Collect all items
Attain 150,000,000 EXP
Max out partner ranks on all characters
Spend over 1,000,000 DP
Clear 10 or more missions with a Z rank
Defeat Kid Buu using an Ultimate Move
Defeat Perfect Cell using an Ultimate Move
Defeat Full Power Frieza using an Ultimate Move
Use all playable characters
Clear all missions
Clear all missions in Special Age
Unlock all characters (excluding downloadable characters)
Clear a difficulty level 5 mission without equipping cards or items
Spend over 100,000 PP
Clear a difficulty level 5 mission without retrying
Clear missions 100 times
Clear the Majin Buu Saga: Majin Buu route
Clear the Majin Buu Saga: Z Fighters route
Clear all missions in Extra Age
Obtain a rarity level 7 card
Attain player rank 99
Clear the Cell Saga: Androids route
Clear the Cell Saga: Z Fighters route
Clear all missions in Another Age
Successfully use an Ultimate Move
Win 100 times in Battle Mode
Acquire 10 gold medals in Battle Royale
Collect 7 Dragon Balls in Dragon Ball Grab
Fight as part of a team in Co-op Mode
Enter a Room in Co-op Mode
Clear the Frieza Saga: Frieza Soldier route
Max out a Meteor Chain
Clear the Frieza Saga: Z Fighters route
Max out a character's partner rank
Clear the Saiyan Saga: Saiyan route
Clear the Saiyan Saga: Z Fighters route
Successfully Chase an enemy
Send ENERGY through an ENERGY Request
Successfully use a Synchronized Rush
Successfully use Revive Soul
Successfully use Energy Share