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