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