Here is the full list of all 45 Broken Age trophies - there are 26 bronze, 16 silver, 2 gold and 1 platinum.
Complete Broken Age
Talk to the boss
Start a group dance party
Make someone strut their stuff
Stand up for someone even though you haven't met her
Stand up for someone even though you haven't met him
Complete Act 2 of Broken Age
Chart a course away from danger
Reconnect with some old friends
Deploy a weapon of mass destruction
Get to know a stranger
Execute a PERFECT escape
Remove 8 sad faces from the Bassinostra
Uncover an unsavory scheme
Get to the ship's brain
Answer some tough questions quickly
Find a genius
Return a missing spaceship part
Get a family member's help on a big project
Find 30 unique reactions to the Spoon
Follow up on a successful performance
Tell a great joke in one attempt
Visit someone in jail
Return a heavy duty spaceship part
Clear up a tangled situation in one attempt
Commission a custom garment
Complete all rescue missions with zero errors
Complete Act 1 of Broken Age
Travel to a forbidden location
Take Control of the Boom Arms
Inhibit a powerful entitiy
Hear 20 reasons why Gary won't grab that
Disrupt Shay's daily routine
Make a Spoon Sweat
Examine all breakfast options
Take down Mog Chothra
Endure a tight situation as long as you can
Feed two different people a healthy snack
Leave an isolated town
Reunite a mother and child
Get to know someone in a precarious position
Escape the Maidens Feast
Learn all you can about Sugar Bunting
Unlock all Broken Age Trophies
Complete Broken Age in under an hour, not counting loading and paused gameplay.