Here is the full list of all 48 Start the Party! trophies - there are 32 bronze, 13 silver, 2 gold and 1 platinum.
Play 10 games of Party Mode
Smash a Piñata in less than 5 seconds
Steal enough stars in the Robber Round to get from last to first place
Steal the last star from someone in the Robber Round, reducing their star count to 0
Bounce an object 10 times without dropping it, during a Three Strikes round
Zap 15+ robots in one game
Survive 20 seconds with a perfect lolly
Catch 10 pizza toppings in one game, during a Three Strikes round
Earn 20 stars in a single Party session
Steal 5 stars in one turn in the Robber Round
Play all the games in Start the Party!™
Get every bonus object within a single Party Mode play session
Munch 5 bugs in one game
Draw on a player’s face
Earn 15 stars in a single Party session
One player wins 3 rounds in a row in Party Mode
Re-record someone's name
Take 100 different player portrait photos
Waft 12 paratroopers to safety
View the credits
Find the 5 bugs in less than 5 seconds
Surrender to 8 angry mobsters
Get to the top of the leader board in Survival Mode
Survive for 2 minutes
Clean all the crocodile's teeth
Play 20 games of Survival
Pop a colour chain of 20+ fish
Get the bonus item in every Party game
Survive for 1 minute 30 seconds
Sshhh, don't disturb the chick!
Completely smash a rock in less than 3 seconds
Survive for 1 minute
Hit 8 moles in a single game
Use 5 litres of hair dye in one game
Play all the Party Games in Free Play
Shoot down 6 ghosts on their way to scaring you
Match 5 characters in one game
Use 5 litres of paint within the templates in one game
Survive for 30 seconds
Swat 50 bugs in one game
Play a game of Start the Party!™ (any game mode) on Christmas Day
Collect all other trophies for Start the Party!™
Pop a colour chain of 70+ fish in Poppin' on Hard difficulty
Play all the Party Games, on all difficulties in Free Play
Play a game of Start the Party!™ (any game mode) between 12 o'clock and 1 o'clock at night
Rescue 3 people just before the monster eats them
Save 30+ chicks in one game
Slice 40 pieces of fruit during a Three Strikes Round