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