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