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