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.
Collect 20 Legendary Weapons
Collect 20 Legendary Outfits
Complete 100 objectives
Craft every Theme
Level up a Dweller from Level 1 to 50
Build 1 of every room type
Scrap 500 items
Collect all Fragments of 4 Theme Recipes
Complete 100 Quests
Craft an Outfit, a Weapon, and a Theme
Upgrade 20 Rooms to Level 3
Collect all Fragments of a Theme Recipe
Make 25 Dialogue Choices on Quests
Successfully stop 50 Raider Attacks
Make 25 Babies
Collect 100 Stimpaks and 100 RadAway on Quests
Change the appearance of 10 Dwellers in the Barbershop
Build 50 rooms
Complete 60 Quests
Level up a dweller from Level 1 to 25
Kill 500 enemies on Quests
Craft 10 Outfits
Complete 30 Quests
Build 25 rooms
Complete 10 Quests
Kill 10 Quest Bosses
Craft 10 Weapons
Collect 10,000 Caps
Successfully complete 50 Room Rushes
Level up a dweller from Level 1 to 10
Merge 3 Water Rooms
Collect 1000 caps
Merge 3 Food Rooms
Merge 3 Power Rooms