There are 51 Trophies that can be collected.

Bronze

Good Fluffy!    
Praise or scold a pet enough times to add a trait.

Sit! Play Dead!    
Earn more than §1,000 during a trick exhibition.     

Have You Seen This Pet?    
Find your lost pet.     

It's Genetic    
Breed a Natural-Born Hunter cat or Natural-Born Digger dog.     

Sim’s Best Friend    
Become a BFF with a dog.     

Wingman, er… Dog    
Fetch a date.     

Where'd You Get That!?    
Fetch something worth at least §2,000 with a dog.     

Do Not Want!    
Successfully bathe your cat.     

Smooth Criminal    
Steal something worth at least §2,000 with a cat.     

Cat-A-Kinesis    
Use a telekinetic cat to make a Sim fix a meal from the fridge.

I'd Rather Be Fishing    
Complete each of the fishing Challenges.

Sunken Treasure    
Find a special plant while fishing.     

Overqualified    
Go to work with 4 career performance requirements at Outstanding.

Mass Hysteria    
Have a dog and a cat become best friends.     

Is This… Okay?    
Make a Sim transmogrified from a cat and a Sim transmogrified from a dog become a couple.     

Family Feud    
Activate the Cry Havoc karma power during a wedding party.     

Maid-out    
Make out with the maid.     

Employee Benefits    
Become a romantic interest of your boss.

Flirting with Disaster    
Escape a karmic backlash by quickly balancing the meter after dropping below -50.     

Yoink!    
With a pet, steal a Sim's clothes who is skinny dipping.     

Dogzilla vs. Cat-Kong    
Destroy §10,000 worth of objects with a dog or cat.     

Leonardo Reborn    
Complete each of the Leonardo Reborn Sub-Challenges.     

Don't Fear the Reaper    
Scare the reaper.     

What Ghosts?    
Catch every ghost spawned by the Ghost Invasion karma power before it ends.     

Bad Neighbor    
Target your neighbor with the Ghostify Power, then catch them with a cat or Ghost Zapper.     

Indubitably!     
Complete each of the mystery chapters.     

The Missing Link    
Discover the lost pirate ship anchor.     

Zombie Bears, Oh My!    
Avoid the zombie bears to find the pirate ship's log.     

Doggie Duties    
Deal with the feline in the Dog Days of Summer mystery.     

Search and Rescue    
Explore the Abandoned Mine in search of Timmy.     

A Punishable Offense    
Punish the Sims for their crumb-filled, sub-standard furniture.     

Don't Talk About Socialite Club    
Get the attention of the Socialite Club.     

Great Scott!    
Create a flux combobulator.     

From the Ground Up     
Build your house from the ground up by replacing your property with an empty lot.

You Da Boss    
Reach level 10 of a career.

Curator's Best Friend    
Complete each of the relic collection Challenges.     

Dancing Fiend    
Complete each of the dancing Challenges.     

Exhibitionist    
Complete each of the exhibitionist Challenges.     

Explorer    
Complete each of the explorer Challenges.

Silver

Personality: I Haz It    
Train a pet to have 9 traits.

Leet Skillz    
Complete any 10 Skill Journal Milestones.     

The Parent Challenge    
Complete each of the parenting Challenges.

No Fate But What We Make    
Send a simbot back in time.

Cat-astrophe    
Complete each of the Cat-astrophe scenario Challenges.     

K9 Cop    
Complete each of the k9 cop scenario Challenges.

TMI    
Have your Sim kiss their brother’s/sister’s best friend.

Mansion Builder    
Build a house with 5 stories and an unfurnished value of at least §100,000.

Must Love Pets    
Promise and complete a pet-related Lifetime Wish.

Gold

All You Need    
Complete each of the love and romance Challenges.

It's Good to be a Sim    
Complete each of the Sim happiness Challenges.

Platinum

Top Dog    
Unlock all other trophies.

http://www.ps3trophies.org/game/the-sims-3-pets/trophies/