This small issue "costed" me some hours of lifetime, so maybe this saves someone else's time:

An otherwise perfectly fine working freshly updated TYPO3 v11.5.x, updated from v10.4.x, threw 404 errors after hitting "login" on the /typo3/ login-page.
Due to circumstances on that web hosting server, I had a hard time finding out that this is a 404 error -- everything having a path like /typo3/path/segement/here resulted in that "not found" thing.
Since in the background the login-page is making an AJAX call, and the address used there also ran in that 404 error, nothing worked.

After searching for that error I found a ticket in track showing the changes needed in the root htaccess file. The .htaccess that worked in v10.4.x needs a few new lines to enable the needed redirects for the backend in v11.5.x. Now where is the/a default root htaccess file for v11?

Turns out it is well hidden in public/typo3/sysext/install/Resources/Private/FolderStructureTemplateFiles/root-htaccess, and it has some other changes compared with the v10.x .htaccess.

So I would suggest to take that file, copy it in the root directory of the TYPO3 installation, rename it to .htaccess and then copy over the custom rules you may have added in the v10.x version.

Peter commented that this is mentioned in the "breaking changes" section of TYPO3 10->11 update preparations: BackendURLRewrites. According to this, only a few lines need to be added/changed, so if you have a large .htaccess with lots of custom rewrites, adding those few lines may be better than to replace the .htaccess with the oriinal v11 one and re-add the custom rules. I think this also depends on how "old" the .htaccess is. I can imagine that if you're upgrading from 6 to 7 to 8 to 9 to 10 and now 11 having the latest .htaccess as a clean base is not so bad.