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