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