(By passenger airline in this question I mean scheduled carriers, with at minimum a few aircraft with 50+ seats.)
I am aware of 3 main classes of airlines:
Are there more? If so, where and how are they used?
Note: I accepted the answer that shows a clear example, but there are many other excellent answers below!
There's also the "Snowbird" model, like Sun Country and Allegiant Air, which primarily fly vacationers and snowbirds between sunbelt (San Diego to Florida) and the Carribean, and the Northern snowbelt states (Maine through Washington State, and Canadian border cities). They are low-cost carriers, with few or no code shares, no connections, no free amenities, and prices that decline as the date gets closer.
Here is Sun Country's route map. Note that nearly all their flights are between Minneapolis (or Dallas) and vacation destinations.
Here is a part of Allegiant's route map, showing cities with flights to Phoenix/Mesa, AZ. They have more routes than are shown here, but all their routes have one end in one of the "sun belt" cities marked in orange.
4. Business class airline
Airline which operate aircraft only have Business class for medium/long haul route. Example: La Compagnie (Paris Gaulle / London Luton - Newark)
Some airline also operated All-Business class service in some of their route. Exapmle: BA Club World London City (London City - JFK), and cancelled Singapore SQ21/22 (Newark - Singapore).
Check Wikipedia for further detail.
Note: Wikipedia claimed that Lufthansa also have All-business class flight but i cannot find such a seat arrangement in seatguru.
PS: SAS have all business-class services between Stavanger and Houston (SK951/952) with b737 operated by Privatair. Qatar also provides all such services between Doha and Heathrow (QR 15/16) with A319 but the service will be terminated soon. (Thanks for @Calchas )
Even within the different categories in the question, there are multiple business models, which can be quite different from each other.
Within low-cost carriers, you have Southwest on one end of the spectrum, which is kind of debatable as to whether it should even be considered a low-cost carrier these days. They charge almost no extra fees for anything and offer frequent flights to almost all major cities, but have base fares that are usually quite comparable to those of the major airlines on the same routes. JetBlue also follows a similar model to Southwest.
On the other end of the spectrum, you have airlines like Allegiant or Wow Air which charge extra fees for just about everything except cabin pressurization, mostly just fly into small-market airports, have very limited route options, offer only a few flights per week on a given route, and often don't offer connections. These are quite different business models.
In the middle of these extremes, you have the business models of airlines like Spirit and Frontier, which have fees and schedules in between those of the airlines on the ends of the spectrum.
Within the regional category, one business model that exists in the U.S. is that of the airlines that exist primarily to provide Essential Air Service (EAS). EAS is a government-subsidized program designed to ensure that scheduled passenger service is available to remote areas that are not near cities large enough to support significant scheduled passenger service profitably on their own. Airlines operating these routes still usually ferry passengers to larger airports where they can connect to other airlines like other regional carriers, but often operate much smaller aircraft (e.g. 13-20-seat aircraft vs. the 50 or 70 seat aircraft common on normal regional carriers.) Airlines exist almost solely to run these types of routes include Great Lakes Airlines and SeaPort Airlines (which ironically has its largest hub in Memphis, which isn't a seaport and doesn't offer service to any seaports.) For examples of the kinds of routes these airlines operate, here's the route map for SeaPort:
Traditional regional carriers within the U.S. typically operate 50-70-seat regional jets. These carriers are usually contracted to 'do business as' the regional wing of one of the major airlines (e.g. Delta Connection, American Eagle, or United Express.) They usually fly from hubs of the airlines they contract with to small-market airports, but not as small as those served by the EAS program. Additionally, the regional carriers are used to provide service between an airline's smaller hubs and medium-market or even large-market airports. For example, in the case of Delta, while a city may have mainline Delta 757 service to Atlanta, it may have only regional airline service to their smaller hubs like MSP, CVG, and SLC.
charge extra fees for just about everything except cabin pressurization
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Yes. The three categories there are not really reflective of many aviation businesses around the globe.
Formerly full service, not yet LCC
In many parts of the world (especially Europe) you have older carriers desperate to reposition themselves as low cost carriers. They have inherited large cost bases and they are trying to shed them, often with limited success. They often adopt the tactics of a "low cost" airline such as charging for amenities and operating on point-to-point routes. But their cost base (per available seat mile) remains as high as even the legacy, global carriers.
So they look like an LCC but actually they would not be considered an LCC by an investor. (Indeed many of them remain owned by sovereign funds.)
In the UK, Flybe (BE) might be a good example, although it is not in government ownership. BE charges low(ish) fares, it focuses on point-to-point routes, most of its routes are under 500 miles, it charges for snacks, it charges for luggage and there is no business class. However, until very recently it had a typical cost per available seat kilometre exceeding that of British Airways shorthaul!
LCCs are Flybe's biggest competitors and their lower average unit cost across their networks allows them to under-price Flybe when they do compete on the same city pairs. Even without direct competition, LCCs have a downward impact on pricing in many markets as they compete for discretionary travel spend. This could become a major threat to Flybe, especially if the level of overlap with LCCs grows.
On the flip side BE offers feed to airlines such as KLM, Cathay Pacific and British Airways. It also wetleased aircraft to Finnair in a joint venture with them called Flybe Nordic (an affiliate member of oneworld). These are not activities typical of a low cost carrier.
I think I would also put CityJet (WX) in this category. It was recently demerged from the AF/KLM group. WX does have a "base" at London City Airport (LCY), but it really focuses on point-to-point rather than transfer traffic (LCY is too expensive to compete for transit traffic). Costs and fares are high, but it's a good airline.
This weak unit revenue trend, at an airport [London City] that is considered to be high yield, is troubling, especially given the capacity cuts. Although the relative trend of RASK [revenue per available seat-kilometre] versus CASK [cost per available seat-kilometre] started to move in the right direction in 2013, it will be a considerable achievement to restore breakeven and then to convert that into sustained profitability.
LCCs now offering full service amenities ("hybrid" carriers)
Many of the original low cost carriers are close to saturating the "low cost" market in Europe and realize the only way to continue growing is to offer fares and service levels that are acceptable to business passengers—or indeed, discerning leisure passengers.
EasyJet (U2) has a new CEO who is making a strong push for business traffic by using central airports, better times, reliability, offering fast lane access, a loyalty programme called "EasyJet plus", and so on. Revenues have grown strongly.
A direct competitor to Flybe on UK domestic routes, whom I use quite often, is a relatively new airline called Eastern Airways (T3). T3 goes the opposite route: luggage is free, all of its passengers get fast track at all airports, it provides a free lounge at its hub, there is free coffee/beer/wine/snacks on board. Yet it charges low fares (most of the time), offers no feed or interline connections, does point-to-point only, and it only operates a fleet of turboprops (and one SAAB).
(However most of their planes have fewer than fifty seats, so maybe this example isn't directly relevant.)
“We’re offering a frequent business schedule with an all-inclusive on board service, both of which are equally important to business travellers.”
http://www.anna.aero/2014/01/22/eastern-airways-returns-onto-leeds-bradford-southampton-route/
All premium long haul carriers
Other posters have mentioned the annoyingly named La Compagnie (try asking "Pardon, où est l'enregistrement pour la Compagnie?" in CDG...) offering all-busines service on PAR-NYC and LON-NYC. There have been one or two of these airlines in the past but they always struggle without the benefit of a major airline to rescue their passengers when the plane goes technical. Odyssey Airlines wants to try the same thing on LCY-JFK, but presently no aircraft can do the route without a stop.
However all of these operations are relatively "niche" and are only operating with about 30-50 seats.
Low cost long haul
Some think this will be the future. The obvious examples are Norwegian (DY) which operates a number of all economy services between Europe and USA and a few between Europe and the far east. I think they deserve to be in a separate group to low cost short haul. The challenges of operating a long haul service are quite different to low cost short haul; on many routes offering only economy and economy plus products is going to be difficult to sustain without sophisticated revenue management work to milk every cent out of those passengers.
Eventually, these operations will require feed from other low cost carriers, they'll require local "hubs" or interchange points, and that's when things will get interesting.
AirAsia X is one example of an airline already going down that route. It has a strong relationship with the various AirAsia brands and offers full interlining between shorthaul and longhaul services. Indeed, the low cost terminal at KUL had to be redesigned to enable bag-collect and re-drop prior to customs/passport control to enable better flow.
Conclusion.
I would agree that many of the airlines I've cited here to rebut your three categories are a little bit exceptional, but I think AirAsia X and EasyJet are the airlines to watch because they are not constrained by starting in one category. The three "niches" you gave will not be hard dividing lines for long.
all-J and all-Y refer to?
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Yet another airline model is local carriers that do not primarily act as spokes, but only for their own niche markets. I believe the planned California Pacific Airlines intends to pursue this model.
Arguably, charter airlines can also be considered another airline business model. Rather than operating on a fixed schedule, these companies will fly on demand. I believe these types of airlines are often used by travel agencies in Europe to ferry vacationers to their destinations. For instance, Germany's Hapag-Lloyd/TUIfly, as well as Germany's former LTU, works this way at least for some of their business.
There are similar small-scale charter operations flying to Southern California's Santa Catalina island, as well as into the Alaska wilderness (and likely also in many other places, such as Yellowstone, Montana)
Another airline business model is specializing on sightseeing tours - flights that depart and land in the same place. LTU used to offer such flights to the North Pole. On a smaller scale, many similar companies exist in or near scenic locations, such as Alaska, Hawaii, the Grand Canyon, and probably also many international locations (Kenya's Serengeti, for instance).
Yet another business model is the private jet shared ownership.
5. "Travel-agent" airline
"Travel-agent" airline stands for airlines owned and operated by the travel agents. It often refers as charter airline, but in this case the lessee and lessor are same company.
The business model of "Travel-agent" airline looks like LCC. They usually provides homogeneous seats; their ticket are only available through their companies; their routes mainly focus on short/medium distance, but with some long tourist cities (like Bangkok) in busy seasons (Summer holiday/Christmas).
On the other hand, they often provide free checked luggage allowance and sometimes free meal. Their seats are generally wider and larger; personal entertainment system may be installed for each passengers.
Thomas Cook airlines and Thomson Airways are tow largest "Travel-agent" airline in UK.
