Earth has Metalbending, Sandbending, and Lavabending.
Water has Bloodbending and Healing.
Fire has Lightning/Electricitybending and Combustion.
What is the subset of Airbending, if there is one?
In Season 3 of The Legend of Korra, the subset of airbending was revealed:
The subset of airbending is flight. This is not the glider-assisted flight that is fairly common, but unsupported flight. It is similar to how Superman or other superheroes fly.
The subset of airbending is unlocked by fully realizing the Airbender Nomad philosophy of removing all earthly attachments. The first airbender to achieve this was Guru Laghima, an airbending master who lived over 3,000 years before the 100 Years War.
Air-bending does not have a subset or amplification mode of its power.
A replica of an airbending scroll by Jeffrey Scott
The Avatar Wiki reports:
Airbending is the only art that apparently has no specific sub-skill or specialized form that has as of yet been revealed, although it has been speculated soundbending (see above) could fit into this category.
Aang has show some unique abilities using his air-bending which could qualify as an amplification but does not include any combat techniques. Again from the Avatar Wiki:
Sound-bending: A specialized form of airbending in which the bender is able to generate massive amplified sound waves with the help of airbending. One possible canon example of this is shown when Aang, while trying to lead animals out of the city to outside the Inner Wall, used airbending to blow his bison whistle, sending massive sound waves throughout the city to call all the animals. He also used it to comically amplify his cry for help when trapped at the Sun Warrior ruins.
One of the possible reasons for a lack of an amplification mode is the defensive nature of the Air-bending style of fighting. Air-benders were pacifists and preferred to avoid conflict rather than engage in it. Many of the amplifications expand the fighting properties of their art significantly and make sense for their development to occur as a practitioner became more skilled in the art. This does not mean there could not be an amplified mode of Air-bending, only that we have not seen one.
While Air-bending does not appear to have an amplified mode, it is one of the most dynamic of the bending styles and offers its practitioners a diverse array of techniques for engaging in and avoiding conflicts. When used by a skilled bender, the air element allows the bender to move faster than the eye can see, increase their combat agility and avoid damage with a fighting style designed to misdirect the opponent, project bursts of air as weapons, create durable shields of moving air, redirect physical objects focused at the bender, and sweep away multiple opponents simultaneously.
Since the series has been restarted with the Legend of Korra, there is always the possibility we will see an amplified mode of airbending.
I personally don't think every form of bending needs subsets and such, it seems to be an expectation of the fans that there be parallels in every branch equivalent to the others, rather than anything stated in the show. As Sokka says during a Korra flashback, some benders just have strange abilities. Bending seems to a very organic landscape.
That being said, the best candidate is probably Spiritual Projection, which is specifically described by Jinora as "a high level airbender technique."
So far she's the only one we've seen do it, (I thought I remembered Aang doing it at some point, but I suppose I'm making that up). However, the way she talks about it, it sounds like it's not terribly difficult to achieve for (unlike Tenzin) spiritually-connected airbending masters.
Airbending is the rarest bending ability and thus has not been fully explored in detail as it was with other bending arts. Plus, most of the knowledge that was present in older generations of airbenders never got passed down to anyone. I think there might have been more, had Firenation not destroyed and burned most of the temples/monks. If there is a subset, it may not have been discovered or it may have been lost.
With the exception of a few Firebenders using fire "jets" to glide short distances, Airbenders are uniquely able to fly under their own power. "Flightbending" perhaps.
Legend of Korra has already shown that technology and science has advanced such that one group of people have learned the physics of aerodynamics with airplanes. Perhaps when the Airbenders/Avatar learn the physics behind flight they can improve upon their stick flyers or to learn to fly without airfoil assistance...
I've seen this debated elsewhere without a concensus, but possibly vacuation (the formation of a vacuum). This may just be an advanced technique rather than a subset.
In Book 3, Episode 10 ('Long Live the Queen') of the Legends of Korra, Queen Hou-Ting of the Earth Kingdom is killed by Zaheer through vacuation. He suffocated her by pulling all the air out of her lungs.
It could be 'pressure-bending', creating a Vacuum around the target or releasing the gases in the body. that would make it the possible deadliest of the specialized techniques.
I could imagine the air-nomades discovering this technique, seeing the horrible impact on the human body and deciding to erase the knowledge from their teachings. it may even be the origin of their passive nature.
If Waterbending has the Deadliest Bending; Waterbending. Maybe Airbending has Oxygen bend. maybe the Air Nomads didn't learn this is because of they're love of peace and freedom. And they apparently had a great sense of humor.
I don't agree that subsets are actually a thing.
I've never agreed with this 'rpg' like approach to bending techniques. All bending techniques for an element are just things someone with that element can learn. Bloodbending isn't metaphysically more special than octopus arms; its just another application of waterbending. Some techniques require special things, true. You can't lightningbend with a clouded heart, void-fly with earthly attachments or bloodbend without a full moon. But that's no different than saying you can't sword fight without a sword. Some techniques rely on certain tools or circumstances. Likewise, some techniques are more difficult to learn but, again, that does not make them special, just hard.
Of course, not everyone will find all techniques easy - even setting aside how difficult they are. Just as some people are better at long jump than high jump or are better than calculus than creative writing, some benders will find some techniques easier than others. In some instances, they may be incapable of a particular technique for all practical purposes (see Bolin and metalbending) but that's because of the way they think and learn, not because they're on a different metaphysical development track.
Fundamentally, and I think this is my thesis, I don't agree that you can pick you one particular technique from a bending school and say, 'this is special. This is a subset.' It comes across as highlight arbitrary.
Certainly their are people who might specialise in a 'subset' (see the metalbending police of Republic City) but that is true about any bending ability. You could have a (hypothetical) Earth Kingdom dojo that is all about using earthbending to sling rocks very fast and very hard. Does that make 'rock throwing' suddenly a subset.
Why is lavabending a subset but not waterbenders ability to change between ice, water and steam, controlling all three?
Why is metalbending a subset when it is literally just controlling the bits of earth left behind in metal?
Take a look at what Iroh said in 'Bitter Work' about lightningbending. "Lightning is a pure expression of firebending without aggression. It is not fueled by rage or emotion the way other firebending is. Some call lightning the cold-blooded fire." Does that sound special or just a cleaver way of using firebending?
The answer is they are not. What is too often labelled a 'subset' is just a slightly rarer use of bog-standard bending. It doesn't have the metaphysical weight to ask why it does or does not exist for airbending.
I would like to posit that Energybending is actually a subset of airbending, and here's why:
Of the 4 Elements of the show, Air is the only one that isn't actually an element in Taoism.
However, Air is equivocal to Energy in Taoism. Qi and Wind are often used interchangeably, and breathing is the method of circulating qi. Because of this, it could be assumed that Energybending is actually the subset of airbending.