The first thing is to search for the serial number in the stolen bike registry. Also use the description, but that can be tough on a common bicycle.

https://bikeindex.org/bikes?stolen=true

Assuming it comes up clean, then take your bike down to the local police office and ask if they have a bicycle registry. If there are problems with the S/N you should deal with it including going to the PD, and have vendor contact info.

If you study/work at a local university, you can often also register the bike at the local campus police office.

Does registering your bike do anything? I'm not sure. Certainly recording your bike's serial number somewhere will help with potential future recovery. Keeping your own photos and S/N may be sufficient for a potential police report, or registering a stolen bicycle online.

Maybe he police sticker will be a deterrent. However, it may also just indicate to the thieves that they should simply part out the bike..