HAT: insult used by the paras to describe non-paras. Apparently, it means Helicopter Assault Trooper, but that might be a backronym, and a backhanded insult to the Paras (whose last combat drop was Suez in 1956). Calling a Para a "hat" to his face is normally considered fighting words.
"Leg" is the US Army equivalent to "hat" - or "dirty, nasty leg" as used by instructors at the airborne school at Fort Benning.
The 82nd and 101st Divisions have a strong rivalry and while the 101st still retains the airborne tab above their unit patch the unit is no longer on jump status and has converted to an air assault (helicopter-borne) structure. Paratroopers in the 82nd will commonly call the 101st troopers disparaging names like "dope on a rope" or "almost airborne" and I've heard the second phrase used to describe other (non-82nd) airborne units on Ft. Bragg such as units in the 18th Corps and Corps Support Command.
US Army airborne wings (awarded after completing 5 jumps in the airborne school) feature a parachute centered with stylized wings behind it. Air Assault wings (awarded after completing the air assault course at Ft. Campbell, home of the 101st) use the same wing structure but replace the parachute in the center with a front view of a UH-1 helicopter. This gives the badge a superficial resemblance to the cartoon character so it is often called the "Bullwinkle Badge."
The army's pathfinder school is located at Ft. Benning as part of the airborne school. Because the Pathfinder badge is the most conspicuous one on the uniform, I've heard the Pathfinder School referred to as the "Badgefinder School."
The US Army has a number of tabs that can be worn above the unit patch if you qualify, usually by completing the appropriate course. The Sapper tab for completing the Sapper Leader Course at Ft. Leonard Wood for example. However, in the Special Operations community only two tabs are thought to matter - the Ranger Tab and the Special Forces Tab. In conversation they are referred to (respectively) as the "short tab" and the "long tab" and a Green Beret (Special Forces) Soldier can be called a "long-tabber."
On a related note, completing the Ranger School and being awarded the Ranger Tab does not make one a "Ranger." I proudly wear the Ranger tab but when someone unknowingly calls me a Ranger I correct them because I have never served in the Ranger Regiment. Ranger units wear a unique patch in the shape of a scroll and it is serving in one of the Ranger battalions that allows you to be called a Ranger. The Ranger unit patch's unique shape lends itself to a tactical procedure used on patrols to quickly cross a linear danger area. When you don't have time to establish detailed security and recon the other side you can "scroll the road" by setting up hasty security teams and sprint everyone across as fast as possible. The hasty security teams face left or right with their left shoulders (where the scroll patch is worn) toward the danger area until everyone else in the patrol passes then they pick up and follow.
One more note about airborne school. If you have completed Ranger school and are not already airborne qualified, you are called a "leg Ranger." Actually, completing Ranger school first is an advantage because you run less risk of injury compared to those who have to jump in each phase of Ranger School. However, showing up to airborne school as a "leg ranger" means you immediately attract extra attention to yourself which is not a good thing. Showing up to airborne school wearing your "Bullwinkle badge" also gets you extra attention. A friend and I did exactly that and spent much of the first week in the "gig pit" with the one "leg ranger" in our jump school class. There we got to "push Ft. Benning" until the instructor got tired. Good times.
And another note on the Ranger tab: Completing Ranger School in the winter is considered harder than the summer and in the old green Class-A uniform, "winter rangers" would sometimes sew their tab on with white thread. Also, whenever you are talking about an Army school with a more recent graduate, it is required that you point out that it was harder when you did it.