A compilation buffer has a lot of underlined links. I would expect
Tab/S-Tab to move between those just as they do in Help or Info (and
in web-browsers ;-).
Should Tab/S-Tab behave the same way in a compilation buffer?
A few days ago I added the [backtab] key binding to the existing [tab]
key binding in grep.el for symmetry. I don't know why [tab] was
bound to `compilation-next-file' instead of `compilation-next-error'.
It is more natural to expect [tab] moving to the next underlined link
(which `compilation-next-error' does). So what about the following patch
(I also deleted commented-out bindings for next-line because this is now
implemented by `next-error-follow-minor-mode'):