As Ken said - for low current traces (CPU data, address lines, etc): 30AWG Kynar insulated wire wrap wire is the standard wire used by most businesses. I still have a few of the small spools that Rat Shack used to sell. We often tack it down with super glue if 1/2" or longer.
We usually use green insulation so it blends in with board. It is solid wire so it conforms to bending, following traces on board and stays put easily.
Small, 50ft spool will last for years for most people.

The Kynar insulation is tough stuff to strip as it tends to stretch so buy a good wire-wrap wire stripper while you are at it. The old click-n-strips worked great but tough to find anylonger.
Another decent stripper came within the Rat Shack wire wrap tools - stored in the handle.
Jameco still has these:
https://www.jameco.com/z/WWT-100-30AWG-Wire-Wrap-Tool_2150361.html

For higher current traces (power transistors, power supplies, etc) - obviously 30AWG won't work. But actual size depends on the current load and there is no easy answer.