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