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