Most debris from a remodel or cleanout can go in your container. Wisconsin's landfill bans and the receiving facility set the limits.
The Wisconsin Department of Natural Resources keeps a list of items banned from landfills. It includes electronics, lead-acid batteries, major appliances, used oil filters, waste oil, and waste tires. Yard waste is on the list as well, including grass clippings, leaves, and brush, and so are common recyclables like cardboard, cans, and bottles.
Hazardous waste, liquids, paint, solvents, fuel, and asbestos can't go in a standard dumpster either. The City points residents to the Milwaukee Metropolitan Sewerage District's household hazardous waste drop-off for chemicals and to appliance recycling services for fridges and freezers.
Ask your account manager about anything you're unsure of before it goes in the container.