Here is the full list of all 51 The Legend of Heroes: Trails of Cold Steel trophies - there are 38 bronze, 10 silver, 2 gold and 1 platinum.
Mastered Trails of Cold Steel and earned every trophy.
Completed the game on nightmare difficulty.
Completed the game on hard or nightmare difficulty.
Completely filled the book section of the notebook.
Forged a strong bond with Towa.
Forged a strong bond with Crow.
Forged a strong bond with Gaius.
Forged a strong bond with Jusis.
Forged a strong bond with Emma.
Forged a strong bond with Machias.
Forged a strong bond with Elliot.
Forged a strong bond with Alisa.
Opened every treasure chest in a single playthrough.
Gathered all 28 varieties of master quartz.
Completely filled the monster guide.
Played for over 100 hours.
Completed all of the character profiles in the notebook.
Won 1000 battles.
Canceled enemy arts and crafts 100 times.
Used Burst in battle 50 times.
Acquired over a million mira.
Completed the final chapter.
Forged a strong bond with Sara.
Forged a strong bond with Millium.
Forged a strong bond with Fie.
Forged a strong bond with Laura.
Achieved student rank A0.
Acquired the strongest weapon for any character.
Completed all quests, both extra-curricular and during field studies.
Obtained all recipes.
Received the Lionheart Medal of Heroism from Principal Vandyck.
Caught all varieties of fish.
Completed Chapter 6.
Won 500 battles.
Initiated 300 advantage encounters (including double and triple advantage).
Raised any two characters to Link Level 5.
Completed Chapter 5.
Raised a master quartz to level 5.
Used over 500 follow-up attacks in battle.
Used Rush in battle 100 times.
Completed Chapter 4.
Completed Chapter 3.
Used 100 S-Breaks in battle.
Completed Chapter 2.
Defeated 8 enemies at once.
Won 100 battles.
Completed Chapter 1.
Earned an S-Rank on any chapter.
Won a battle with a tactical bonus of x2.0 or greater.
Completed the prologue.
Arrived in Trista for the first day of school.