Here is the full list of all 21 Barbie and Her Sisters Puppy Rescue (EU) trophies - there are 4 bronze, 8 silver, 8 gold and 1 platinum.
Earn all of the trophies in the game
Remove fleas from 15 puppies
Help 20 puppies clear the obstacle course
Finish trick training 10 times
Vaccinate 10 puppies
Play with 25 different puppies
Rescue 25 puppies
Brush the teeth of 20 puppies
Adopt your own puppy
Guide a puppy out of the storm drains
Find a puppy in the hedge maze
Find a perfect home for a puppy
Bathe, blow dry, and brush a puppy
Earn the trust of a puppy in hiding
Clip a puppy's nails without making a mistake
Teach a puppy to "stay"
Play with a puppy in the Kennels
Find a puppy by following its paw prints
Name a stray puppy
Complete a rescue mission
Give dental care to a puppy