There are 38 Trophies that can be collected.

Bronze

Dive For Five
Save 5 Divers Simultaneously in Super Sub Rescue

Trigger Happy!
Destroy 5 waves of UFOs without the use of the charged shot in Space Shootout

Heli Hero
Save 3 Civilians simultaneously in Chopper Rescue

Flying Fish
Catch 10 fish in mid-air in a single game of Deep Sea Peril

Triple Blast
Destroy 3 UFOs with one charged blast in Robo Blast

Jelly Justice
Pop 10 jellyfish within a single game of Gone Fishin'

Firefly Feast
Grab 40 or more fireflies within a single game of Jurassic Spark

Top Row Rescue
Catch 10 Sailors using only the top row of windows in a single game of Overboard

Production Line
Fix 20 robots within a single game of Robo Repair

Party Hero
Play every game within Start the Party!™ Save the World

Highspeed Hero
Complete a game of Quickfire

House Party
Complete a game of Party mode

Silver

Packed Like Sardines
Save 14 Divers in a single game of Super Sub Rescue

Supermassive Black Hole
Lure 4 UFOs into a Black Hole in Space Shootout

Dinner Denied
Prevent the monster from eating any Civilians in a single game of Chopper Rescue

Fish Fever
Catch 40 fish within a single game of Deep Sea Peril

Civilian Saviour
Protect every Civilian in a single game of Robo Blast

Frenzied Fishin'
Rescue every diver over 8 areas within a single game of Gone Fishin'

Quick Fix
Fix every robot within a single game of Fixin' Frenzy

Tyrannosaurus Wreck
Protect all the cavemen in a single game of Jurassic Spark

Line em' Up
Destroy 8 UFOs simultaneously in Alien Attack

Home Sweet Home
Help 6 cavemen get back home within a single game of Caveman Bounce

Quite the Catch
Catch 25 Sailors in a row in a single game of Overboard

Manic Mechanic
Repair every robot within a single game of Robo Repair

There's No Space Like Home
Help 10 Astronauts get safely to their rocket in Astro Bounce

Boom Shot!
Shoot 5 bombs in a single game of Speedy Shootout

Ready For Launch
Build 6 rockets in a single game of Rocket Repair

Trail Blazing
Destroy a group of 9 Fireflies in one stroke in Fire Fight

Safe N' Sound
Rescue 25 divers from the clutches of the Octosquid in a single game of Tentacle Trouble

Flash Bash
Don't let the bears crack the window in a game of Bear Bash

Bomb Bounder
Destroy 35 of Dr. Terrible's cloned helicopters in a single game of Bomb Bouncin'

Medical Marvel
Collect 20 medi-packs within a single game of Manic Medic

Survival Expert
Climb to the top of the Highscore Table in Survivor mode

Clear Winner
Get a top score in any game in Free Play mode

Party Marathon
Complete a 'Long' game of Party mode play

Gold

Survivor
Survive for 4 minutes or more in Survivor mode

Highscore Legend!
Play every game in Free Play mode and get onto the Highscore Table in each one

Platinum

Party King!
Collect all other trophies for Start the Party!™ Save the World

http://www.ps3trophies.com/forums/ps3-trophy-news/6914-start-party-save-world-trophy-list.html