Ordinary building and cleanout debris is fine. The limits come from state law, from the City, and from the facility that receives the load.
The state building standards code sets the headline duty on a permitted project here, and it reaches the customer rather than the hauler. An applicable construction or demolition project has to divert at least 65 percent of its nonhazardous debris from landfill through recycling or reuse, separate recyclable and reusable material such as wood, metal, concrete, asphalt, cardboard and drywall, and keep receipts and weight tickets showing where everything went. That makes sorted containers an advantage rather than a chore on a Beverly Hills job.
Hazardous waste, liquids, paint, solvents, fuel, used oil and asbestos cannot go in a standard container. California keeps household hazardous material out of the trash altogether, which covers batteries, fluorescent tubes, electronics, thermostats and similar items.
The City's own bulky item collection excludes construction material and hazardous waste by name, so none of it can be left at a curb for pickup either. Tires, propane cylinders and appliances holding refrigerant should be set aside and raised with your account manager when you book.