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
Completed all 62 side quests
Obtained 66 armor or weapons
Cleared all 52 dungeons
Completed the Main Quest
Obtained all 7 skills
Cleared 25 dungeons
Reached level 50
Defeated the Arcane Dragon
Located all monuments
Obtained 30 armor or weapons
Defeated the Ice Dragon
Completed 30 side quests
Defeated the Flame Dragon
Located half the monuments
Obtained 4 skills
Reached level 10
Completed 6 side quests
Cleared 5 dungeons
Obtained 2 skills
Obtained 3 armor or weapons
Located one monument