Most "regular" troops don't get a handgun at all. They get M16s, M4s, M249 SAWs, and M240s (G or B). Some older units may still have M60s, but the 240 is replacing them. Oh, and can't forget our grenadiers with the M203 attch to an M4 or M16. :)
Typically, you will only see handguns with medics, officers, senior NCOs (and some platoon sgts), and machine gunners (M240s/M60s). Other folks you might see with one are combat vehicle crewmembers (CVC) and aviation personnel.
To your original question, the M9 is the primary sidearm, but we also have the M11 (Sig 228) for compacts. Tankers/CVCs and aviation personnel, as well as AFOSI, Army CID and NCIS are issued these.
Taking personal weapons is strictly prohibited, not only by your unit, and your service (Army, Navy, AF, Marines), but also by the Joint Forces Commander in theater. Do some folks do it and get away with it? Occasionally, but it isn't worth it. The "any NATO" rule doesn't exist either. The military is pretty strict about you carrying a weapon you are qualified with only. If I have not qualified on the M203, I don't get to carry one. If I am not qualified with the BHP or CZ75, I can't carry them either - even if they are NATO approved.