Here is the full list of all 29 Trigonarium achievements.
Complete Original Arcade World 1
Complete Original Arcade mode (single player or coop)
Complete Ultimate Arcade mode (single player or coop)
Complete Original Arcade mode without dying (single player or coop)
Complete Ultimate Arcade mode without dying (single player or coop)
Complete Ultimate Arcade World 1
Complete Original Arcade World 2
Complete Ultimate Arcade World 2
Complete Original Arcade World 3
Complete Ultimate Arcade World 3
Complete Ultimate Arcade World 4
Complete Ultimate Arcade World 5
Shoot 10000 bullets
Boost 100 times
Destroy 10000 enemies
Score 35 million points in Original Arcade mode
Score 500 million points in Ultimate Arcade mode
Score 5 million points in The Worm in Original single level mode
Score 50 million points in The Worm in Ultimate single level mode
Score 10 million points in The Eye in Original single level mode
Score 50 million points in The Eye in Ultimate single level mode
Score 14 million points in The Spirit in Original single level mode
Score 70 million points in The Spirit in Ultimate single level mode
Score 100 million points in The Scarab in Ultimate single level mode
Score 200 million points in The Mothership in Ultimate single level mode
Score 2 million points in Quick Attack mode
Score 4 million points in Chase mode
Score 3 million points in Classic mode
Score 1.5 million points in Super mode