How can I browse the web on very old versions of Mac OS?
This answer was last updated in February 2021, and may become outdated as time goes on. As of this writing, however, all software listed is actively maintained with security patches. On OS's which themselves have known vulnerabilities, this is particularly crucial.

Mac OS 8 and below
Attempting to browse the web on these OS's is basically a lost cause. However, you could set up Web Rendering Proxy on another (modern) machine, which will stream websites to older browsers as an interactive gif. If you squint, it's almost like the real thing! https://github.com/tenox7/wrp

Mac OS 9 – Mac OS X 10.3
Also more-or-less a lost cause, but if you want to try your luck, go with Classilla. http://www.floodgap.com/software/classilla/. You might also consider Web Rendering Proxy (see above).

Mac OS X 10.4 &quot;Tiger&quot; &amp; 10.5 &quot;Leopard&quot;
Cameron Kaiser's TenFourFox is here for you! This browser was forked from a very old version of Firefox, so expect to run into broken websites on occasion. For casual browsing on a secondary computer, however, TenFourFox is perfectly sufficient.
PPC users should download builds from the official website: http://www.floodgap.com/software/tenfourfox/
Intel builds are also available at https://sourceforge.net/projects/tenfourfox/files/unstable/contrib/. These are considered &quot;unofficial&quot;, and the author requests that Intel users do not report bugs or ask for support.

Mac OS X 10.6 &quot;Snow Leopard&quot;
InterWeb
With support for WebRTC and an engine based on Firefox 52, Wicknix's InterWeb browser is likely to offer the best compatibility with modern websites on Snow Leopard. Make sure to download the 10.6-specific version: https://randommacstuff.blogspot.com/p/interweb-browser.html
Arctic Fox
If InterWeb runs too slowly on your old hardware, consider trying this other browser from the same developer. It's based on an older engine which won't work with as many websites, but it should be a bit more performant. https://github.com/wicknix/Arctic-Fox/wiki

Mac OS X 10.7 &quot;Lion&quot; &amp; 10.8 &quot;Mountain Lion&quot;
This is the real deal! Blueboxd maintains Chromium Legacy, an up-to-date copy of Chromium (the open source variant of Google Chrome) modified to be compatible with Lion and Mountain Lion. It will work with every website Google Chrome does! https://github.com/blueboxd/chromium-legacy/releases
I have created a &quot;Chromium Legacy Downloader&quot; Preference Pane that makes it a bit easier to install updates: https://jonathanalland.com/downloads/chromium-legacy-downloader.zip

OS X 10.9 &quot;Mavericks&quot;
My favorite version of OS X, and personal daily driver! The best option for Mavericks is Firefox 78 ESR, which is still officially supported by Mozilla with automatic security updates! https://support.mozilla.org/en-US/kb/switch-to-firefox-extended-support-release-esr
Alternately, if you would prefer a Chromium-based browser, Chromium Legacy (see above) is also now compatible with Mavericks.