Firefox family browsers.
These have good compatibility because Chrome, Firefox, Edge and Safari are the browsers most commonly tested by web developers. With more obscure browsers, many commercial website operators like to blame the user for breakage and refuse to fix issues unless you switch to one of their "approved" browsers. Firefox is common enough that they cannot get away with refusing to support it.
Firefox itself is not particularly privacy friendly by default. Arguably, this is a consequence of significant turnover in the Mozilla foundation, as many important figures have left years ago. Companies with a questionable record like Google are also major funders of Mozilla and possibly exercise malicious influence.
However, Firefox still exposes many useful configuration options that can greatly enhance privacy and user control if changed from their defaults. These are all in
about:config, and although official docs are not great, the online community is usually helpful at understanding them. It is possible to set these using
user.js so you can simply synchronize that file across all your computers to have unified settings without having to use Firefox Sync. A few others (like default extensions and search engines) can be configured with policies, which are also quite powerful.
It's worth noting that configuration has become slightly restricted over the years. For example, there used to be straightforward keys for disabling the auto update, which were then removed. The more obscure methods, like blanking the update URL, were also eventually removed. However, even so, Firefox still has much more configurability than other major browsers.
There are many extensions for Firefox, including very useful ones like uBlock Origin, decentraleyes, cookie autodelete, vimium, redirector, downthemall. Originally, Firefox had a lot of theming and extension capabilities, but these have been restricted in recent years (with the switch to WebExtensions), which has led to some extensions becoming abandonware and it became more difficult to create your own extensions.
Many "improved" variants of Firefox exist, which come with some settings improved over the defaults. These come and go, so it is wise to not become too dependent on them. Librewolf is a notable one. Librewolf has many useful customizations that enhance privacy and security. However, it also has many useless changes that are security theater and sometimes harm usability with no real benefit, like disabling the password manager and making it harder to re-enable it. The salient changes of Librewolf are available as open source. Librewolf also lags behind new Firefox versions, but this is not useful for avoiding undesirable updates from Firefox because Librewolf still catches up eventually.
I would say the best option is to review the changes in Librewolf, and then decide how many of them matter to you. If most matter, install Librewolf and disable ones you don't want. If most or half don't matter, simply replicate them in Firefox Developer Edition yourself. Librewolf is overzealous in certain things like DRM and it can break websites that rely on that, Librewolf devs' attitude is that "you shouldn't use those sites anyway".
Other notable variants:
- Tor Browser is based on Firefox. It is not practical to use Firefox for Tor these days, because they've diverged considerably and it creates a fingerprinting risk, but the good news is that many privacy improvements were introduced by Tor devs and added back to the Firefox codebase.
- Waterfox originally created because Firefox was not 64 bit and had poor performance. This has since changed. Waterfox also has some privacy enhancements, but these can be replicated in Firefox more easily.