Here is the full list of all 19 Picross Fairytale: Legend of the Mermaid achievements.
All Trophies collected
Tutorial finished
Solved in less, then 60 seconds
All levels with 3 stars
Island world complete
All mini-games complete
All siren voices collected
Sunken Ship world complete
Colorful Reef world complete
Earned 100.000 coins
50 hints used
100 levels with 3 stars
Mermaid's Cave world complete
20 hints bought
Mini-game complete
World completed without hints
Atlantis world complete
40 levels with 3 stars
3 levels in raw without hints