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