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