what are common engine problems with the 77 buick skylark

Age with a '77. That's about it. From what I could find, and going back to 1977 there is not a lot of information/owners reviews but both the 5.0 (302) V8 and the V6 3.8 nobody had a bad thing to say.

There were quite a few engine options available, Buick, Olds, Pontiac, and Chevy engines could be had in a Skylark, the only "common problem" you might find fault with is how most GM engines of that era were redesigned circa mid-70's in response to the energy crunch and thus had emissions equipment on them that was more of an annoyance than anything else. If you're looking at one, check the subframe (where it meets the body) and engine mounts for rot, like you would any car of that age.