Here is the full list of all 35 Fallout Shelter trophies - there are 20 bronze, 10 silver, 4 gold and 1 platinum.
Change the appearance of 10 Dwellers in the Barbershop
Complete 60 Quests
Craft an Outfit, a Weapon, and a Theme
Collect all Fragments of 4 Theme Recipes
Make 25 Babies
Collect all Fragments of a Theme Recipe
Collect 100 Stimpaks and 100 RadAway on Quests
Level up a dweller from Level 1 to 25
Complete 30 Quests
Craft 10 Outfits
Make 25 Dialogue Choices on Quests
Upgrade 20 Rooms to Level 3
Build 50 rooms
Kill 500 enemies on Quests
Complete 10 Quests
Craft 10 Weapons
Kill 10 Quest Bosses
Build 25 rooms
Successfully complete 50 Room Rushes
Collect 10,000 Caps
Level up a dweller from Level 1 to 10
Collect 1000 caps
Merge 3 Water Rooms
Merge 3 Power Rooms
Merge 3 Food Rooms
Collect all other 34 Trophies for this Trophy.
Level up a Dweller from Level 1 to 50
Build 1 of every room type
Collect 20 Legendary Weapons
Collect 20 Legendary Outfits
Successfully stop 50 Raider Attacks
Complete 100 objectives
Complete 100 Quests
Craft every Theme
Scrap 500 items