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