How to earn the Garbage Day achievement / trophy
Kill the Garbage Collector and acquire the Fugue slab. Go to the new door on the eastern cliff near the entrance of Karl's Bay Afternoon. Go inside and defeat the Garbage Man. Pick up his Slab ability.
How to get the Garbage Disposal Facility trinket
Under the grinder on the conveyor belt is a new unique trinket. To destroy the grinder we need to blow it up with three grenades, or alternatively feed it three items that it doesn't like - e.g. Turrets, Crankwheels, etc.