Tampermonkey is, by a considerable margin, the most popular userscript manager. It runs as an extension on Chrome, Microsoft Edge, Safari, Opera Next, and Firefox. If you prefer open source projects, then there's
Violentmonkey, which is available on Chrome, Opera, many other Chromium based browsers, Maxthon, and Firefox. Greasemonkey 4 (GM4) is also open source and works on Firefox.