Nearly all construction and cleanout debris can go in your container. Michigan's landfill rules set the main limits.
Michigan keeps a list of landfill-prohibited materials from the Department of Environment, Great Lakes, and Energy. It includes beverage containers with a deposit, yard clippings, empty drums, lead-acid batteries, used oil, bulk liquids, medical waste, hazardous waste, and scrap tires. Asbestos and a few other special wastes need extra handling before any landfill will take them.
Two items on that list trip people up. Grass, leaves, sticks, and landscape trimmings should be composted, not landfilled. Whole tires are also a problem, since a landfill can only take them after they've been cut or shredded.
Paint, solvents, fuel, propane tanks, and other hazardous materials don't belong in a standard dumpster either.
If you have a question about a specific item, your account manager can help.