Star Trek actually has a LOT of types of FTL travel. In fact, due to the tendency for some of the series' to have "tech du jour" I would argue it is a top contender for "most" forms of FTL travel. I have listed the ones I can think of off the top of my head, but I am sure there are more!
Although some of the technologies mentioned have similar names (for instance, a lot of them have "warp" in their names) they often have differences which are significant enough to count them as different technologies. Often, the word "warp" simply refers to speed (anything FTL), even though warp drive or bending of space may not be involved.
(0). I write this only to bring Star Trek on par with HGTTG, which claims the Improbability Drive, which takes you to every point in the universe simultaneously. In the infamous episode Threshold, (which I understand now the writers disavow writing, even though technically it's canon!) Tom Paris pilots a shuttlecraft which apparently uses some form of multispectral quantum subspace engine design to travel at infinite velocity, allowing him to be at every point in the universe simultaneously.
Warp Drive, obviously, but there were several types of warp. In the original series, there was no visible warp effect, while by the movies there was a kind of rainbow trail when a ship went to warp. In TNG and beyond, vessels going to warp generated a kind of while flash. In general, anything that uses a warp coil to generate a subspace field geometry is warp drive.
Trans-warp
The word "warp" appears to be deceptive here, in that the term is a kind of catch-all for several technologies which are merely "faster than warp". The movies introduced transwarp developed by the federation, but this was later abandoned, and it bears little resemblance to other things of the same name. In TNG and beyond the Borg have trans-warp, and although it uses something called a "trans-warp coil", it appears to operate differently than regular warp drive, and even requires a network of trans-warp conduits to use. Sometimes these conduits are said to be created by the ship itself, but in others (for instance, Endgame) it is implied they exist separately, for instance to the existence of hubs, and when for example the ship has to choose which conduit to take to get to an aperture.
Coaxial Warp
Seems to still work by folding space, but appears to do it by changing the geometries of subspace particles.
Wormhole
A tunnel connecting two points in spacetime via subspace. There are actually several different way of creating wormholes which may distinguish them as separate kinds of travel:
a. "Normal" wormholes which are naturally occurring
b. The Bajoran wormhole was constructed by the Prophets using verterons
c. In 2372 the Federation experimented with a technique involving generating a strong subspace tensor matrix, and then sending out a magneton pulse which would produce or interact with a tetryon field to generate the wormhole.
d. Warp engine imbalances sometimes create temporary wormholes.
e. An interspatial flexure was another type of wormhole.
f. A Möbius inversion was a type of wormhole with level-six subspace distortions and gravimetric shear.
g. The MIDAS array was able to create a short-lived micro-wormhole by firing a tachyon beam at a pulsar.
Spatial flexure
A spatial rift that could be generated by a standard deflector array.
A Spatial Vortex
Also similar in appearance to a wormhole, but different in manifesting as an actual physical passageway between two points in space.
Quantum Slipstream Drive
Said to be similar to transwarp, but apparently didn't need the network of conduits.
The method of travel used in Where No One Has Gone Before to increase warp drive efficiency and travel to the realm of thought. This method appears to use the mind or thought to affect the travel.
The Trajector in Voyager where a whole planet was used to send people across the galaxy via folding space.
Dimension Shifts appear to make FTL speed possible as they can fold space.
The Nexus allowed one to travel to any point in time and space instantly (making it FTL) although since being in the Nexus removed all motivation to do so, and the only recorded instance of it occuring is when Jean Luc Picard used it to travel back to stop the destruction of the Veridian system.
As @NPSF3000 and @Dan pointed out, The Soliton Wave was a wave phenomena that travelled faster than light which a ship could "ride".
@MartianInvader reminded me of Iconian gates. At least two types of Iconian technology provided FTL travel. The first, the Iconian Gateways, allowed for instantaneous travel for at least 70,000 light years. The second was a device hybridized by the Romulans from Iconian technology which allowed the Hobus supernova to travel at multi-warp speeds.
As @Izkata mentioned, The Geodesic Fold worked by targeting the magnetic field of a red giant star with a verteron beam.
@Iskata also pointed out the Subspace corridor, which was distinct from the Borg transwarp conduit.
@apsillers point out the Graviton Catapult operated using a graviton surge that propelled the ship into null space.
Also mentioned by @apsillers was the Displacement wave, which was used by the Caretaker to transport Voyager from the alpha quadrant to the delta quadrant in a short period of time. This method of FTL travel has similarities with the technology used by the Guardian of Forever.
The Xindi subspace vortex created an energy vortex using a phase deflector pulse and could transport a vessel dozens of light years in a matter of hours.
Also brought up by @Izkata, the Subspace distortion used by Reginald Barclay while under the influence of the Cytherian probe. This technology used an emanation of a highly-charged graviton field from the vessel's warp nacelles, and the resulting field transported the Enterprise to the center of the galaxy in only a few seconds.
The Energy Vortex was temporal continuum anomaly similar in nature to a tractor beam but which also possessed intelligence. This method was used for FTL travel once, when Captain Jean Luc Picard travelled through it some distance and six hours into the past, reversing his internal chronometer in the process.
The Subspace transporter used subspace instead of space to transport matter faster than light.
Transwarp beaming combined the transporter with transwarp theory.
The Nyrian Translocator combined transporter technology with a spatial distortion field.
Temporal transporters achieved FTL travel by combining transporter technology with temporal mechanics by some kind of utilization of chroniton flux.
As @Damon points out, Bajorans built Tachyon Light ships which used solar sails to catch tachyon eddies at least 800 years prior to Deep Space Nine's arrival.
The Chrono deflector operated by generating a powerful tachyon pulse to form a temporal rift, which could then be directed to a specific time and place. Admiral Janeway used this device in 2404 to travel from the alpha quadrant to the delta quadrant in a matter of seconds.
The Aldean Repulsor Beam was able to push the Enterprise D far enough in an instant that it took three days to get back at maximum warp.
Temporal Displacement Drive was a 31st century technology that used temporal radiation to travel through time and space at faster than the speed of light.
Ion propulsion was used for FTL travel.
The Lang Cycle Fusion drive was also an FTL technology.
The Crystalline Entity travelled FTL by chanelling collected electromagnetic energy and organic matter converted to energy through its crystalline structure.
Tin Man aka Gomtuu used some kind of energy wave to travel (or repel other ships) FTL.
Various other alien entities had FTL capability which was never explained but seemed to be either biological or achieved without any obvious technology:
a. The method used by the Douwd to propel the fake Husnock ships at FTL speeds to either chase the Enterprise away or retreat.
b. The changeling Laas transforms into a spaceship which moves with a "pulsating motion" like "swimming".
c. Q's method of travel when chasing the Enterprise. This method seemed to be limited to about Warp 9.9
d. Q's method of travel in which he popped in and out, which had a similar appearance going to Warp in TNG. However, this same effect also accompanied other actions such as creating things out of nothing or dispatching them, and it is unlikely it was the same form of travel.
e. The method by which Q transported the Enterprise to the delta quadrant. This was distinctly different from (e) as it was much faster, and (f) which had a different manifestation.
Along the same lines, any device in the Star Trek universe that allowed time travel could also be considered an FTL device. There are many instances of this, such as the Guardian of Forever (although see #16), the 29th Century timeships, and the time pod stolen by Berlinghoff Rasmussen.
Arguably, the method used to pull parallel universe enterprises into the current universe was FTL, as those ships would all have to be outside the light cone of the main enterprise. This could also apply to other cases where travel to other universes (such as the mirror universe) occurred.
