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
Brush the teeth of 20 puppies
Help 20 puppies clear the obstacle course
Remove fleas from 15 puppies
Play with 25 different puppies
Finish trick training 10 times
Vaccinate 10 puppies
Rescue 25 puppies
Adopt your own puppy
Find a puppy by following its paw prints
Bathe, blow dry, and brush a puppy
Earn the trust of a puppy in hiding
Teach a puppy to "stay"
Find a perfect home for a puppy
Clip a puppy's nails without making a mistake
Play with a puppy in the Kennels
Name a stray puppy
Guide a puppy out of the storm drains
Complete a rescue mission
Give dental care to a puppy
Find a puppy in the hedge maze