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