Learn English – More grammatically correct: “anything but” or “anything except”

I'd say that there is a subtle distinction between the two, in that "correct" implies that the given answer (or answers) is definitive and absolute, as might be the case, for example, with a simple maths problem.

In contrast, "right" would be better for an answer which, although considered appropriate, is still matter of opinion (even if most people agree). This might apply, for example, to a question of ethics ("is this the right thing to do?"); again, using "correct" here would imply that there is an absolute answer that no-one could possibly disagree with.

"Right" can still be used in places where "correct" would be appropriate (such as a maths problem), but not the necessarily the other way around.

I have been using the following phrases but I am still not confident that they are grammatically correct and sound right:

"in regards with something"

"in regard to" is the right way here.

"with regards to something"

This is OK. Somehow I have the feeling that "with regard to" is more normal though. Paul Brians seems to back this up.

"regarding something"

This is OK, e.g. film title "Regarding Henry", etc.

I have also heard/read people using an arbitrary combination of the above (e.g. "in regards to"). Are those correct? If yes - are they equivalent or the usage depends on context?

It's much easier to answer your question if you put in some examples you've seen. Anyway I hope the above is enough.

Note that there is a completely different usage "regards to" as in "give my regards to old Broadway and tell them I will soon be there" etc.