I really think you are going to need video clips due to terminology differences.
In the Moo Sul Kwan system of taekwondo, they define a
jumping kick as a kick where you jump off the same leg you kick with
(for example, jump front kick with right leg, start with right leg in front, pull left knee up in their air to initiate jump and then kick with right leg)
and
flying kick as a kick where you jump off the opposite leg of kicking leg
(for example, a typical flying side kick — like in Chong Moo, for example —you start right leg in back, step forward with right leg, left leg steps forward and springs you up and forward, then tucks, while right leg side kicks)
So, by these definitions, you start with basic:
jump front
jump roundhouse
jump hook (more difficult, IMO, than jump spinning hook or flying hook)
jump side (IMO, a VERY difficult jump kick)
flying front
flying roundhouse
flying hook
flying side (the aerial easiest aerial side kick, I think most people would agree)
I would then suggest maybe:
jump turning back kick (jumping off both legs)
jump spinning hook kick (jumping off both legs)
jump spinning inside crescent (the one that uses the outside edge of foot as weapon, jumping off both legs)
jump spinning outside crescent (the one that uses the inside edge of the foot as weapon — this kick often called the "butterfly" kick)
and when you are feeling REALLY froggy,
jump spinning wheel kick/jump spinning heel kick (jumping off both legs)
Then you could do some of the still basic, but not AS basic:
jump stomp (front thrusting heel)
flying stomp
jump arc
flying arc
jump axe (inside and outside — these are NOT as hard as they sound to do)
(don't think I've ever tried axe kicks as flying kick ... hmmmm)
jump twist
jump spinning twist (off both legs)
flying twist
Then if you REALLY want to push yourself, there are multiple kicks:
jump front-round
jump double side (like flying side, but bottom leg also kicks instead of tucks)
jump double front -- both feet close together, hitting same target
jump double front "split" kick -- feet far apart, hitting different targets
flying side-backside (I think it is Best of the Best or maybe Best of the Best 2 that shows the guy with the eye patch doing this one ... we used to train this one on the heavy bag.. very fun)
flying side-spinning hook (my little brother SCORED on me with this one once while we were free sparring ... was simply beautiful)
flying side-spinning heel (not sure if I've ever SEEN this one done ...)
triple front (right-left-right)
double front-roundhouse (we used to do this one on the heavy bag, too: right front kick+left front kick+right roundhouse)
flying side-roundhouse (hitting bag/target simultaneously: the bottom foot kicks like a double flying side, but top foot is roundhouse instead of side)
oh yea: and jump spinning heel kick but instead of jumping off BOTH legs, jump off just the leg your are kicking with: so left leg swings forward, then jump off right and kick with right. If you can pull this one off, you can even develop a multiple kick: jump front-spinning heel, for example
jump double back kick (probably have to land on your hands and feet, but if you can actually land on your feet after this one, more power to ya!)
I'm starting to sound like Forest Gump's friend, Bubba, who introduced him to all the different shrimp dishes
While I started out mentioning the MSK TKD definition of jumping/flying kicking, as I understand it the jumping/flying kicking in our system actually has roots in the Musulkwan hapkido school Grandmaster Lee H. Park named his American schools after. Nearly all kicks were "borrowed" or appropriated from hapkido for taekwondo, which some people will tell you actually started out with only a couple kicks -- front, side, round.
In any case, the (Korean) Musulkwan was renknowned for its flying (jumping and flying) kicking.
The last thing I'm going to write is: I am NOT recommending you try ANY of these. Doing so is at your OWN risk. These types of advanced techniques should really only be attempted under the guidance of a qualified instructor.
As pointed out above, there is a real risk of knee injury if do something wrong*— or even if you are doing them RIGHT and just get fatigued and make a mistake!
And that's all I have to say about that
