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