Handlebar on my hybrids, stem on the road bike. I use mine for traffic documentation. If I'm stopped I can aim the camera toward a potential problem. That's harder to do with the camera mounted on the head tube or elsewhere on the frame.

However the handlebar mounted camera can't readily be aimed while moving, unless I physically move the camera on the ballhead mount. But I need to crank the ballhead mount very tightly to prevent creeping so it's a compromise. Usually I leave the camera/mount in one position.

Image stability depends on the road. On smooth pavement it's fine. Not too bad on our crushed limestone MUP and most passable gravel roads. It's worst on chipseal and striated concrete. I doubt any rigid mount could ensure vibration-free videos on chipseal. I'm pretty sure my camera has built in vibration reduction, although I don't know whether it's sensor based, lens based or software. There's a slight jello effect in videos recorded on really rough pavement, but better than YouTube's optional software stabilization, which often produces a hallucinogenic jello warp drive effect.

There are spring loaded or shock absorbing type mounts designed to reduce vibration, although the main effect appears to be protecting electronics from vibration damage, not improving video stability. A Steadicam type floating mount would need to be impractically bulky for normal bike riding.

Some of the most serious commuters who feel the need to record their rides have front and rear bike mounted cams and helmet cams. This seems to be more common in urban areas with streets that are too narrow for the heavy traffic volume and/or vehicle speed. Some cities in England seem to have this problem, and some YouTubers post their worst encounters, demonstrating various angles from two or three cameras.

For recording fine details in typical US license plates you need 60 fps or faster, and either bright enough light to force a fast shutter speed, or good enough low light sensitivity and wide dynamic range to record details at night. My Ion Speed Pro is fine in daylight but mediocre at night. Noise and poor dynamic range obscure fine detail, so all I can usually record is the shape, color and general image of a car. The license plates are usually blown out featureless white because Texas plates are reflective white.

I don't know if any action cams can do much better. We'd need a Sony one-inch type sensor for better nighttime performance with lower noise and better dynamic range. I don't see that happening anytime soon. Most action cams still use compromised teensy sensors, which do offer the advantage of depth of field to record more from near to far in reasonably sharp focus. And some dash cams have good low light performance, although they aren't designed for bicycles and aren't weatherproof as-is.