Here is the full list of all 44 Harry Potter & The Half-Blood Prince trophies - there are 32 bronze, 7 silver, 4 gold and 1 platinum.
Earned all trophies
Clean up after Fred and George at The Burrow
View Dumbledore's first memory of Tom Riddle in the Pensieve
Assemble the Gryffindor Quidditch Team
Create a great Christmas party for Slughorn
Protect The Burrow
Help the student retrieve her stolen Gobstone
Help Ron avoid Lavender
Hide the Half-Blood Prince's Potions Book
Retrieve Slughorn’s missing Memory
Win the Quidditch Cup
Escape the Cave
Used your first portrait shortcut
Use all of the portrait shortcuts
Conclude the adventure
Complete Crest Collector Level 1
Complete Crest Collector Level 2
Complete Crest Collector Level 3
Complete Crest Collector Level 4
Complete Crest Collector Level 5
Complete Crest Collector Level 6
Complete Crest Collector Level 7
Complete Crest Collector Level 8
Complete Crest Collector Level 9
Complete Crest Collector Level 10
Complete Crest Collector Level 11
Complete Crest Collector Level 12
Complete Crest Collector Level 13
Complete Crest Collector Level 14
Earn the Potions Beginner's Luck Badge
Earn the Speedy Brewer Badge
Earn the Potions Club Star Badge
Earn the Potions Club Veteran Badge
Earn the Master Potioneer Badge
Earn the Duelling Beginner's Luck Badge
Earn the Reflex Badge
Earn the Duelling Club Champion Badge
Earn the Duelling Veteran Badge
Earned the Master Duellist Badge
Earn the Quidditch Beginner's Luck Badge
Earn the Precision Flyer Badge
Earn the Combo Flyer Badge
Earn the Keen Flyer Badge
Earn the Master of Flying Badge