Ordinary construction and cleanout debris loads without issue: lumber, drywall, shingles, siding, flooring, cabinets, doors, carpet, furniture and household junk.
Hazardous household material does not. Paint, stains, solvents, fuel, motor oil, antifreeze, garden chemicals and batteries all belong at the county drop-off described below, and city rules separately bar depositing any of that on a street, sidewalk or trail.
Electronics come out of the load. Televisions and monitors are handled and charged separately in Dakota County, as are tires, so pull them before you start filling.
Yard waste has its own route in Eagan. Grass, leaves, brush and prunings are kept apart from refuse by the city's own definitions and go to composting rather than into a mixed container.
Asbestos never belongs in a standard container. Houses from Eagan's big 1960s and 1970s building waves can hide it in floor tile, pipe wrap and some board products, so if you uncover something questionable during demolition, stop and call your account manager before the load goes any further.