Here is the full list of all 42 Puyo Puyo Tetris achievements.
Clear 50
Played all 5 Arcade modes
Cleared 10,000
Won your first Puzzle League match
Defeated 10 opponents in an Endurance match
Defeated 15 opponents in an Endurance match
Performed a 3-Chain 100 times in completed matches
Popped 10,000
Cleared 3 different-colored
Cleared 1,000
Won your first Free Play match
Defeated 5 opponents in an Endurance match
Defeated 3 opponents in an Endurance match
Popped 1,000
Played all 3 Puyo Puyo Challenge modes
Played all 3 Tetris Challenge modes
Complete Adventure
50% adventure mode
70% adventure mode
Won ten Puzzle League matches
100% adventure mode
Performed a 4-Chain 100 times in completed matches
Performed a 5-Chain 100 times in completed matches
Performed a 6-Chain 100 times in completed matches
Performed an All Clear 50 times in completed matches
Performed an All Clear 100 times in completed matches
Cleared 4 different-colored
Popped 50,000
Performed a 7-Chain 150 times in completed matches
Performed an All Clear 150 times in completed matches
Popped 100,000
Clear 100
Performed a Back-to-back 25 times in competed matches
Performed a Back-to-back 50 times in competed matches
5-Combo
8-Combo
perfect clear
perfect clear 3 times
Cleared 50,000
Clear 200
Performed a Back-to-back 100 times in competed matches
Cleared 100,000