Ordinary construction and household debris is fine. The limits come from Michigan's landfill rules and from what the receiving facility will take.
Michigan keeps several things out of landfills statewide. Yard clippings, including grass, leaves, sticks, and trimmings, have to be composted instead. Whole tires, lead acid batteries, and used oil are all banned, and so are bulk liquids unless they've been solidified first. Empty drums have to be crushed or filled with solid waste before a landfill can take them.
Hazardous waste, paint, solvents, fuel, and asbestos never belong in a roll-off container.
The transfer station adds rules of its own. It takes no yard waste at all, and none of the authority's sites accept brush, branches, or woody tree material right now. Royal Oak residents can bring household hazardous waste like oil-based paint, motor oil, pesticides, and fluorescent bulbs to the drop-off center on Coolidge in Troy, by appointment and with proof of residency.
Got an item you can't place? Ask your account manager before it goes in the container.