Here is the full list of all 35 Fallout Shelter trophies - there are 20 bronze, 10 silver, 4 gold and 1 platinum.
Upgrade 20 Rooms to Level 3
Build 50 rooms
Merge 3 Water Rooms
Merge 3 Food Rooms
Merge 3 Power Rooms
Build 25 rooms
Collect 10,000 Caps
Level up a dweller from Level 1 to 10
Collect 1000 caps
Collect all other 34 Trophies for this Trophy.
Successfully complete 50 Room Rushes
Level up a dweller from Level 1 to 25
Level up a Dweller from Level 1 to 50
Build 1 of every room type
Collect 20 Legendary Weapons
Collect 20 Legendary Outfits
Successfully stop 50 Raider Attacks
Complete 100 objectives
Make 25 Babies
Make 25 Dialogue Choices on Quests
Kill 10 Quest Bosses
Complete 10 Quests
Complete 30 Quests
Complete 60 Quests
Complete 100 Quests
Collect all Fragments of a Theme Recipe
Collect all Fragments of 4 Theme Recipes
Craft every Theme
Kill 500 enemies on Quests
Collect 100 Stimpaks and 100 RadAway on Quests
Scrap 500 items
Craft 10 Weapons
Craft 10 Outfits
Craft an Outfit, a Weapon, and a Theme
Change the appearance of 10 Dwellers in the Barbershop