Most debris from cleanouts and remodels can go in your container. The limits come from Michigan's landfill rules and from the landfill that takes your load.
Michigan's Department of Environment, Great Lakes, and Energy bars several materials from regular landfills. The list includes asbestos, hazardous waste, car batteries, used oil, bulk liquids, medical waste, PCB waste, and yard clippings. Scrap tires can only be landfilled after they're cut up or shredded.
Hazardous waste, liquids, paint, solvents, and fuel can't go in a standard dumpster either. If you're putting out an old fridge or freezer for curbside bulk pickup instead, the City's hauler needs the doors off and won't take anything that still holds Freon.
When you're unsure about an item, ask your account manager before you load it.