That's an excellent and thoughtful question about something most of us use daily without a second thought. I'm happy to report that in most modern cities, the answer is a reassuring no. Contemporary urban trash cans are specifically designed with public safety as a top priority.
You'll find that the vast majority of new public litter bins feature smooth, rounded edges and curved contours. Manufacturers and city planners have moved away from sharp, angular designs, specifically to prevent the kind of scratches, torn clothing, or more serious injuries you're concerned about. The materials themselves also contribute to safety; many are made from thick, molded plastics or coated metals that don't develop sharp burrs or rusted edges over time.
The focus is on creating a piece of public infrastructure that is not only functional but also passively safe for everyone, including children running in a park or a visually impaired person navigating a sidewalk. So, while it's always wise to be cautious in public spaces, you can generally feel confident that the humble trash can has been engineered with your well-being in mind.