That's a great question! Getting rid of a broken trash bin responsibly is more important than many people realize. First, don't just leave it on the curb with your regular trash, as most municipal collection services won't take it that way. Your best first step is to contact your local city or county waste management authority directly. Many municipalities offer specific "bulky item" pickup days or will provide instructions for scheduling a special collection for large, non-standard items like a broken bin. Often, this service is free or has a minimal fee.
If the bin is made of plastic (most are) and is reasonably clean, check if your local recycling center accepts it. Call ahead or check their website for "rigid plastics" acceptance. Some centers have specific drop-off areas for large plastic items. If the plastic is recyclable, they will typically process it. For metal bins, recycling is almost always the best and easiest path.
If the bin is beyond repair and not accepted for recycling, then municipal bulky waste pickup is your most responsible option. This ensures it goes to the proper processing facility. Avoid illegal dumping at all costs—it's harmful to the environment and often carries heavy fines. A little bit of research into your local services is the key to disposing of your old bin the right way.