An interesting one right now is the use of the word "Jew". It kind of occupies a murky grey area in US parlance right now. It's an ok word to use but it makes many uncomfortable or squeamish. You can see people kind of grimace when you use it. Generally Jewish or Jewish Person is considered more acceptable.
This is true in the midwestern U.S., too. For example, if I'm being asked about the religion of a Jewish deli owner, the question will almost always be, "Is he really Jewish?", not "Is he really a Jew?" Whereas, "Is he really a Lithuanian?" would be just as acceptable as "Is he really Lithuanian?"
"Happy holidays" is a good politically correct example. "Winter break/winter holiday", along the same vein, is fairly common. I'm a bit confused by JollyRoger's post. In the midwestern U.S., "Happy holidays" is the politically correct term, and "Merry Christmas" is either the common informal term or the already-know-the-audience-is-Christian term. "Holiday party" for a festive gathering is also used to be politically correct, though you do still have Christmas parties by Christians (whether practicing or culturally), as well as by non-Christians who think the political correctness is silly.
"Socially challenged" is another fairly common one to refer to someone who is socially inept. This would be used to describe people who are breaking social norms due to being oblivious to them when most people would be aware of them, not to describe people who like to spend lots of time alone (also known as "loners"). "Loners" could simply like to spend time by themselves but be perfectly socially capable when they so chose.
"Mentally challenged" is along the same line, and "challenged" in general is a common politically correct modifier. Ex. "vertically challenged" for short people.
Politically correct terms for fat people include "well-built", "muscular" (if they have muscles as well as general mass), and the feminine-only "curvy". These can also be used to describe people who aren't fat, so it somewhat depends on context. More politically correct than "fat" but probably less so than the previous would include "hefty".
I feel like there's probably a politically correct term for "hillbilly" and "redneck", but the closest I can come up with are referring to them by their places of residence (ex. "residents of Appalachia", the over-inclusive "West Virginians" (who don't all qualify as hillbillies, but the state has that reputation), etc.). Again, these would depend on context - if you ask "What's he like?" about someone you don't know, and the first reply is, "Oh, you know, he's from West Virginia" or "Oh, you know, he's a NASCAR fan", it might be hinting at them being a hillbilly/redneck. But if the reply is, "Oh, he's a cool guy, we went to college together, occasionally took a trip to a NASCAR race" then NASCAR is probably just one of his interests, and he's not necessarily a redneck. But someone else might know a better general-purpose politically correct term here.
Also, in the midwestern U.S., "Asian" is not particularly politically correct. It's not necessarily offensive, however, and it's still used commonly. If a restaurant serves Chinese, Indian, and Thai food, no one's going to object to calling it "Asian". But "Asian" is tricky, because a lot of time Americans use it because they don't know enough about Asia to be more specific. "Is she Korean?" is a fair question. "Is she Asian?" implies you are unaware of the vast differences in culture in Asia, although it's still more politically correct than "Is she oriental?" "Where is she from?" would be a more politically correct way to ask if you really don't know if she's from Korea, Japan, or China, for example. Similarly, people probably won't look at you funny if you recount a story about "some Asian guy" you encountered any more than they would if it were "some Chinese guy". (Also, in the U.S., "Asian" generally refers to Chinese, Japanese, Korean, and to a lesser extent southeastern Asian in this informal context)