That's a great and very practical question! The answer depends entirely on your specific city or municipality's public works department. Many cities do have some form of replacement program for publicly owned, damaged urban trash cans, but it's rarely called a formal "trade-in" like you'd find with a phone or car.
Typically, if a city-issued trash can (the large, public bins on sidewalks or in parks) is broken, vandalized, or worn out, you should report it to your local public works or sanitation department. They will assess the damage and usually replace it at no direct cost to you, as maintaining public infrastructure is part of their service. The process is more of a service request than a trade-in with value exchange.
For residential bins provided by the city for curbside pickup, policies vary. Some municipalities offer free replacements for damaged bins, while others may charge a fee. It's less common to "trade in" an old one for a discount on a new one; they usually just collect the broken bin and deliver a new one.
Your best course of action is to visit your city's official website and look for the "Sanitation" or "Public Works" section. Search for "trash can replacement" or "report damaged bin." You can often submit an online request or find a direct phone number. Be ready to provide the location (e.g., cross streets for public bins or your address for residential ones) and a brief description of the damage.
So, while there might not be a commercial trade-in program, an efficient replacement system usually exists. A quick call or online report is your first step to getting that damaged bin swapped out.