Here is the full list of all 23 Cat Quest trophies - there are 7 bronze, 7 silver, 8 gold and 1 platinum.
MEOW!
Reached level 99
Obtained 66 armor or weapons
Cleared all 52 dungeons
Completed the Main Quest
Completed all 62 side quests
Obtained all 7 skills
Defeated the Arcane Dragon
Cleared 25 dungeons
Located all monuments
Reached level 50
Defeated the Ice Dragon
Obtained 30 armor or weapons
Completed 30 side quests
Obtained 4 skills
Located half the monuments
Defeated the Flame Dragon
Obtained 2 skills
Cleared 5 dungeons
Completed 6 side quests
Reached level 10
Obtained 3 armor or weapons
Located one monument