Short answer: there is no single universal rating, but a typical well-maintained urban bench can handle roughly 400 to 600 pounds of static, evenly distributed weight before structural failure becomes a real concern. But wobbling is not really about total capacity. You will often feel movement at 150 to 250 pounds if someone sits heavily on one end, shifts around, or drops onto the seat.
The wobble usually comes from the connection points and installation, not from the seat material itself. Over time, bolts loosen, mounting plates pull slightly away from pavement, legs settle unevenly, brackets rust, or wooden slats shrink. Any of those issues can make a bench rock with one person even when the steel or wood itself is still strong. Lightweight metal benches may flex under a single adult because the frames are built to be inexpensive and easy to move. Concrete benches rarely wobble unless the ground underneath has settled or cracked.
So in practical terms, most urban benches will not fail under normal use, but the wobble threshold is often around 200 to 300 pounds of uneven or shifting load, not the full weight limit. If a bench rocks when a normal adult sits down, it usually needs tightening, re-leveling, or new anchors rather than replacement. The safest way to judge a specific bench is to ask how tight the hardware is and how level the feet are, not just how much weight the seat can hold.