There are 43 Trophies that can be collected.

Bronze

Smarty Branches    
Grow the Tree of Wisdom to at least 100 feet    

Crash of the Titan    
Defeat your first Gargantuar    

Don't Pea in the Pool    
Complete a daytime pool level with using Pea-shooters of any kind.

Soil Your Plants    
Plant your first 10 peashooters

Explodonator    
Blast 10 zombies at once with a Cherry Bomb!    

Close Shave    
Survive the zombie onslaught after all lawnmowers have been used.

Nom Nom Nom    
Stop the horde using only the Sunflower, Wall-Nur, and Chomper on any level.

Penny Pincher    
Pick up 30 coins in a row on a single level without letting any disappear.    

Sunny Days    
Accumulate 8,000 sun during a single level.    

Popcorn Party     
Defeat 2 Gargantuars with Corn Cob missiles in a single level.    

Good Morning    
Complete a daytime level by planting only Mushrooms and Coffee Beans.

Last Mown Standing    
Defeat the last zombie in a level with a lawn mower.    

20 Below Zero    
Immobilize 20 full-sized zombies with a single Ice-shroom.    

Flower Power    
Keep 10 Twin Sunflowers alive in a single level.    

Pyromaniac    
Complete a level using only explosive plants to kill zombies.    

Lawn Mower Man    
Kill 10 zombies with a single lawn mower.    

Chill Out    
Feel the rhythm, feel the rhyme, you've one level to destroy 3 bobsleds, its Jalapeno time!    

Defcorn 5    
Build 5 Cob Cannons in a single level.    

Monster Mash    
Crush 5 zombies with a single Squash.    

Blind Faith    
Complete an extremely foggy level with using Planterns or Blovers.

Melon-y Lane    
Plant a Winter Melon on every lane.

Enlightened    
Collect all Zen Garden, Mushroom Garden and Aquarium Garden plants.    

Diamond Beghouler    
Upgrade all your plants in Beghouled    

Greenfingers    
Grow 10 Zen Garden plants to full size    

Wall-Not Attack    
Complete ZomBotany without using Wall-nuts, Tall-nuts or Pumpkins

Sol Invictus    
Complete Last Stand with 2,000 sun remaining    

Sproing! Sproing!    
Complete Pogo Party without using Tall-Nuts or Magnet-shrooms.     

SPUDOW!    
Blow up a zombie using a Potato Mine.

Silver

Home Lawn Security    
Comlete adventure mode    

Master of Morticulture    
Collect all 49 plants in the game

Alive and Planting    
Survive 40 waves of pure zombie ferocity

Roll Some Heads    
Bowl over 5 zombies with a single Wall-nut.

Shopaholic    
Go trunk driving and spend at least $25,000 on Crazy Dave's Twiddydinkies.

Grounded    
Defeat a normal roof level without using any Catapult plants.    

Zombologist    
Discover the Yeti zombie.

No Fungus Among Us    
Complete a nighttime level without planting any Mushrooms.

Pool's Closed    
Complete a pool level without using water plants.

Lucky Spin    
Get 3 diamonds in Slot Machine    

Chili Free    
Complete Column Like You See'em without using Jalapenos

Shooting Star    
Complete Seeing Stars before the first Flag Zombie appears

Gold

Second Life    
Complete Adventure mode a second time.

Beyond the Grave    
Beat all 20 mini games

Platinum

The Complete Zombie    
You've won every trophy, climbed every mountain, defeated every zombie, planted every plant. Congratulations!

http://www.ps3trophies.org/game/plants-vs-zombies-vita/trophies/