Players can freely choose anything (or almost anything - some campaigns may ban some options) from the PHB and, if we're using a published campaign setting, the main book for that setting. Oh, but no multiclassing.
Players can then use anything from one additional official sourcebook. (Exception: in almost all cases, they can't use options from a setting book unless we're actually using that setting. Exceptions would include Spelljammer or Planescape, which obviously have much more of an "anything goes" vibe than FR or Eberron.)
Depending on the campaign, there may be a small number of additional options that are either homebrewed or taken from third-party supplements. But these are few and far between. And the PHB+1 rule still applies - so you can't use both something from Tasha's and the Gunslinger subclass (or whatever).
One last thing: I haven't actually told my players that they can use materials beyond the PHB, nor do I intend to do so. If and when someone asks, the answer will be as above... but I'm still waiting on them doing so.