Here is the full list of all 19 KOI trophies - there are 17 bronze and 2 silver.
Open all the flowers in the game
Collect all the Stars in the game
Collect all the Puzzle Pieces in the game
Unlock all the Skins
Complete every level
Dodge 20 attacks of the Black Fish
Reach the frontier of the 3rd part of Level 7
Remain undiscovered by the Black Fish in Levels 2 and 5
Suffer 20 attacks by the Black Fish
Complete the upstream part of Level 7 in under 3 minutes
Complete the upstream part of Level 7 in over 3 minutes
Don't miss a single encounter with Old Toad
Lead more than 50 small fish in total
Bump into an obstacle 200 times
Total distance traveled over 8000 meters
Total play time is 24 hours
Make an opened flower light up 30 times
Get hit by sparks 5 times
Avoid being hit by a single spark in Level 6