From my experience, Tampermonkey is best suited for Chromium-based browsers such as Google Chrome and Opera.

Violentmonkey is currently the best supported userscript manager for Firefox.

Tampermonkey

With more than 10 million users, Tampermonkey is the leading userscript manager. It's also available in all the major navigators.

Here are the direct pages where you can install the Tampermonkey extension:

More detailed instructions to install Tampermonkey and a comprehensive features overview is presented in the following links: Google Chrome, Opera, Microsoft Edge, Safari.

Violentmonkey

Violentmonkey is the main alternative to Tampermonkey, and is completely open source. It's available in Firefox, Chrome, Opera and Maxthon.

The following page lists the download links for the different supported browsers: https://violentmonkey.github.io/get-it/

Greasemonkey

Greasemonkey is the original userscript manager. Following Firefox 57, and starting from the version Greasemonkey 4, the extension changed its API to be compatible with the Browser Extension API, and no longer support scripts written using the older GM_ synchronous API (more explanations can be found in this blog post).

I discourage using it at the moment because the majority of user scripts are written using the older API and the other userscript managers don't support the new asynchronous API.