Most debris from a remodel, a roof job, or a cleanout can go right in. The limits come from Michigan's landfill rules and from the landfill that takes your load.
Michigan keeps a list of materials its landfills can't accept. It includes yard clippings like grass, leaves, and brush, and whole tires unless they're cut in half. Car batteries and other lead-acid batteries are on it, along with used oil and bottles or cans that carry a deposit. Empty drums have to be crushed first, and asbestos falls under its own federal handling rules.
The City's voucher page adds the local landfill's own list. It won't take liquids, hazardous waste, fluorescent light ballasts, anything containing PCBs, radioactive materials, medical waste such as syringes, asbestos, or drums. It also turns away mixed loads with yard waste in them.
For your dumpster, that means no hazardous waste, liquids, paint, solvents, fuel, or asbestos. For tires, old TVs, and refrigerators, Bay County points residents to drop-off sites such as Bay City Recycles on North Water Street in Bay City.
If you're not sure about an item, ask your account manager before you load it.