Most construction and cleanout debris is exactly what a container here is for, since the city will not take it any other way. The limits come from the city's own rules and from what the landfill accepts.
Kentucky does not publish a statewide list of banned materials the way some states do. What the state and the city do is fix the destination: disposing of waste anywhere inside Frankfort other than at a licensed and permitted landfill or transfer station is prohibited. Burying debris on a job site is not an option.
Nothing hazardous belongs in a standard container: no liquids, no wet paint, no solvents, no fuel. The city makes the paint point directly, asking that leftovers be solidified with sand, dirt, or cat litter rather than poured off. Dead animals and petroleum products are barred from waste containers outright.
Electronics have their own route and should come out of the load. Tires, appliances, and shingles are all treated as awkward material locally, so mention them when you book.
Loads also have to be tarped on the way to the landfill, so nothing should be mounded above the rails. If an item has you guessing, ask your account manager before it goes in.