Ordinary construction and cleanout debris is fine: lumber, drywall, shingles, siding, flooring, cabinets, doors, carpet, furniture and household junk.
Used oil is handled separately and by name. The city makes it unlawful to dispose of used oil except through a used oil collector, to mix used oil with solid waste going to a landfill, or to pour it on the ground or into a drain.
Wet paint is out. The city bars disposing of wet paint outside the state environmental agency's requirements, so paint has to be dried out or taken to a collection point.
Lead acid batteries are barred from public disposal facilities under Utah law, and tires, televisions and monitors are all handled and priced on their own at the district sites, so pull them before you load.
Appliances holding refrigerant carry an extraction fee at the district, so flag a fridge or an air conditioner when you book rather than letting it ride in the container.
Asbestos never belongs in a standard container. If older material turns up on a remodel, stop and call your account manager so the right handling can be arranged.