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