Here is the full list of all 10 No Time to Explain trophies - there are 4 bronze, 5 silver and 1 gold.
Beat the scrolling shooter segment without dying, as a dinosaur
Finish Shotgun universe in under 60 seconds
Dance through the first universe and don't stop for nothing
Collect all the hats
Finish the whole damn game
Collect half the hats
Start playing
Play through Straitjacket Darkness universe without dying, with a horse head
Play through Fat Cake universe and eat every fat cake, with a hamburger for a face
Beat Rejected Green Clone universe in under a minute, wearing a monocle and a top-hat