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