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