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.
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.