There are several methods of faster-than-light (and even some faster-than-warp) travel that have been studied or encountered by Starfleet personnel. Most of the technologies listed here are either experimental or unavailable.
Soliton Wave Drive
Created by Doctor Ja'Dar, the soliton wave system offered the promise of a unique new technology for the Federation - warp speed travel without warp drive. The system involved a set of field generators located on or near a planet which would generate a self-sustaining subspace field wave. This would push a vessel into warp speed, carrying it in a straight line at constant speed. At the other end of the journey another group of field generators would disperse the wave, causing the vessel to drop back out of warp.
The benefits of the system would have been enormous. The drive can only be used between star systems where there is already a set of field generators in place, and so would not have replaced warp drive equipped ships in their exploratory roles. However, all traffic between already settled systems could have been accomplished via vessels with no warp drive, vessels which would therefore have been far easier to construct and maintain. In addition, the soliton system is 450% more efficient than a normal starship warp drive, offering considerable energy savings.
Unfortunately, the first test of the soliton wave was only partially successful. After some time at warp speeds the unmanned test vehicle was destroyed by shear forces. The wave began to accelerate and grow in size and power, to the extent that it would destroy most of the planet when it arrived at its destination. Fortunately the Enterprise-D, which was monitoring the test, was able to to disperse the wave using a photon torpedo salvo. The Federation was continuing research into this system, but as of yet there is no reported success.
Null Space Catapult
In 2376 Voyager encountered an alien named Tash who had constructed what he termed a "null space catapult". Like Voyager, Tash had become lost far from home. In his case he had been exploring an unstable wormhole when it threw his ship across thousands of light years, leaving him with a journey of ten years to get home. Rather than engage in such a voyage Tash built the catapult.
The device consisted of a large framework on which were mounted eight graviton projectors powered by a tetrion reactor. A ship would station itself close to the catapult and allow itself to be encased in graviton surge, which would then catapult it into null space. Mr Tash said that the ship should emerge a few hours later hundreds or even thousands of light years away. When Voyager arrived he was having some difficulty with the catapult - a test conducted earlier had been partially successful, throwing a probe some six hundred light years away. Unfortunately, the catapults core had been destabilized in the test and Tash had been unable to fix it. Voyagers captain offered to help by using well timed graviton bursts from the ships main deflector to correct the problem. In return, Tash offered to allow Voyager to use the catapult once he was gone.
Tash was able to use the catapult shortly afterwards, achieving results well beyond his expectations; his ship was thrown five thousand light years closer to home. He had to adjust his shield system in mid flight as the stresses involved in using the catapult almost tore off the outer layer of his ships hull, but he was able to transmit the modifications to Voyager and the ship used the catapult itself safely. Voyager was thrown across thirty sectors of space, coming three thousand light years closer to home in less than one hour.
Voyager has only encountered one other species which uses tetrion technology, the Ocampans Caretaker which kidnapped the ship in the first place. The rarity of the technology indicates that it is unlikely that Starfleet will be able to construct another catapult or similar technology in the near future, if at all.
(Quantum) Slipstream Drive
In mid 2374 Voyager received an encoded message from Starfleet via an alien communications system. The crew were unable to decode the message, and so it was eventually filed and more or less forgotten about.
At the end of the year Voyager encountered Arturis, an alien with an exceptional gift for languages who decoded the message for Janeway. The crew was directed to a set of co-ordinates, at which they discovered a Starfleet vessel named the USS Dauntless, NX 01-A. The message told them that the Dauntless had been launched on Stardate 51472, covering the 60,000 light year distance to Voyager in less than three months - an average speed of 240,000 times that of light. The ship was unmanned, and more than large enough to accommodate Voyagers entire crew.
On boarding the ship Voyagers crew found that it was powered by a "quantum slipstream drive" rather than a matter / antimatter reactor. The drive core was much more compact than a typical Starships power core. It directed energy into both the two nacelles and the main deflector dish, with the latter system creating the slipstreaming effect. The slipstream is somewhat similar to the transwarp drive conduits used by the Borg, but can be generated by any Starfleet vessel with the appropriate modifications to its navigational deflector. Unfortunately, the quantum stresses on the hull are such that a normal vessel such as Voyager cannot survive in a slipstream for more than an hour or so.
It transpired that the Dauntless itself was actually a fake created by Arturis. His species had been assimilated by the Borg shortly after Janeway ended their war with Species 8472 and he blamed her and her crew for assisting the Borg during that conflict. He had created the Dauntless using particle synthesis, an advanced technology which appeared to function in a manner related to holodeck imagery. His plan was to lure the crew onto the ship and then send it into Borg space on autopilot so that the Borg would assimilate them all. Although Voyagers crew became suspicious of Arturis, he managed to abduct Janeway and Seven of Nine and escape into slipstream. Voyager followed and recovered its crew members, leaving Arturis to continue on to Borg space alone. Voyager managed to get 300 light years closer to home before its slipstream collapsed.
Voyagers crew continued to work on the slipstream concept, and several months later managed to come up with a modified version of the technology. The new slipstream drive used a quantum matrix combined with benamite crystals and Borg technology to achieve far higher speeds than the Dauntless had managed, well in excess of a billion times the speed of light. Unfortunately, before the flight Lieutenant Paris discovered a previously unnoticed 0.42 phase variance in the slipstream threshold. Such a variance was more than enough to cause a catastrophic failure in the slipstream, leading to the destruction of the ship.
Captain Janeway nevertheless attempted the flight, sending the Delta Flyer ahead of the ship to map the slipstreams phase variations and send corrections back to the ship. The approach did not work, and Voyager was indeed thrown out of the slipstream after travelling only 10,000 light years closer to home, albeit without major damage. It was subsequently discovered that the phase corrections sent to the ship had in fact originated in an alternate timeline in which the ship had been destroyed during the flight. Surviving members of the crew had sent the information from decades in the future in order to rewrite history and save Voyager from destruction.
The benamite crystals used in the drive were highly unstable, and once they had been used up the flight could not be repeated. Voyagers crew continued to work on this impressive technology in the hope that further progress could be made.
Co-axial Warp Drive
On Stardate 51762 Voyager encountered Steth, an alien being who was testing a form of drive system based on a co-axial warp core. Starfleet engineers had dreamed about this system for years, but had never been able to make a working example. The system allows a vessel to 'fold' the very fabric of space, allowing a ship to travel instantaneously across large distances.
The ship Voyager encountered was in some difficulty, with the co-axial warp core threatening to explode - an event which would cause all space within one billion kilometres to collapse. Voyager was able to correct the problem by encasing the ship in a symmetric warp field to contain any instabilities in the core, a technique covered in Advanced subspace geometry.
The ships pilot, Steth, said he had jumped to Voyagers location from the Benthan system some twenty light years away. He asked for help in repairing the drive and Captain Janeway agreed. The problem lay in the co-axial induction drive, a unit which draws in subatomic particles and reconfigures their internal geometry in order to make space folding possible. Particle instabilities would overload the ships engines; after some investigation, Lieutenant Paris decided that they needed to dilute the particle stream as it entered the drive using a polaric modulator from one of Voyagers impulse engines. This would act in a manner similar to a 20th century automobile, providing a quick and easy solution to the problem.
With this modification the co-axial drive performed apparently flawlessly, and although Voyager suffered some problems due to Steth's antisocial actions while on board the ship, eventually the vessel proceeded on its way.
Underspace corridors
The underspace corridor network surrounding the Vaadwuar homeworld is a natural phenomenon which allows ships to travel at far greater than normal speeds. The corridors were formerly controlled by the Vaadwuar themselves, who spent centuries mapping them and used them to allow trade with many other species and for military expansion of their territory. Approximately one thousand years ago the Vaadwuar were defeated by an alliance composed of their various victim species including the Turei. The alliance subsequently claimed control of the corridor network.
In 2376 Voyager was caught up in the network. The ship had been cruising at warp six when one of the corridors pulled them in abruptly, attracting the ships warp field like a magnet. They found hundreds of interconnected corridors, most filled with debris of various kinds. The ship found it difficult to escape from the corridors, but a Turei vessel was able to use a resonance pulse to alter Voyagers shield harmonic and expel it from the corridor.
Voyager found that the corridor had allowed them to cross two hundred light years of space in under five minutes, an average speed of over twenty million times the speed of light. They attempted to negotiate the right to use the corridors further, but were unable to do so.
It is unknown if similar corridors exist anywhere in Federation territory.
Methods not requiring a ship
Iconian gateway
Until 2365, within the Federation it was widely believed that the Iconians were a myth. They were widely known in archaeological circles as ancient 'demons of air and darkness' who were able to appear at will anywhere, travelling without benefit of spacecraft. On Stardate 42597.17 Captain Donald Varley of the Galaxy class USS Yamato was given an artefact thought to be Iconian when visiting an archaeological site. He was able to translate symbols on the artefact and after correcting for 200,000 years of stellar drift he was able to determine that the location of Iconia itself was in the Romulan Neutral Zone, close to Romulan space.
Varley proceeded into the Zone and encountered an active piece of Iconian technology around Stardate 42607.95. The device scanned his ship, whereupon the Yamato began to suffer various computer malfunctions. Immobilized, Varley called Captain Picard for assistance. Unfortunately, shortly after Picard arrived the Yamato began an emergency dump of its antimatter stocks. The dump was then halted and the containment fields dropped while sufficient antimatter was still present to cause a catastrophic explosion.
The Enterprise itself then began to experience similar computer malfunctions. It was determined that the Yamato had contracted a form of computer virus from the Iconian probe, which had communicated itself to the Enterprise via the communications system when the ship had transmitted its log entries. The Enterprise was able to destroy another probe before it could cause further contamination.
On Iconia itself Picard discovered the source of many of the legends concerning the Iconians; the Iconian gateway. The gate appeared as an oblong panel in the air with various destinations displayed in it. To reach a destination one simply stepped through at the proper time, crossing light years in a moment.
Picard was able to use the gateway to escape the planet after setting the generator equipment to self destruct so as to prevent it from falling into Romulan hands. The ship was able to clear out the Iconian program by conducting a complete shutdown and reboot of all computer systems and escaped from the area without further incident.
A second Iconian gateway was discovered in the Gamma Quadrant by a force of rebel Jem'Hadar, who planned to use it to break away from the Founders and overthrow the Dominion. A joint Federation / Dominion assault force raided the location of the gate, which was protected by neutronium armour impenetrable to conventional weapons and a dampening field which rendered phasers inoperable. They joint force were able to fight their way to the gate and plant explosives to destroy it.
To date no further trace of functional Iconian technology has been discovered.
Sikarian trajector
Encountered by Voyager in 2371, the Sikarians are a highly advanced and peaceful people who are legendary in the Delta Quadrant for their generosity and kindness to strangers. Their technology is in general in advance of that of the Federation, most especially when it comes to transporters. The Sikarians used a device which they called the trajector, a transporter capable of sending people or objects across distances of 40,000 light years in an instant. The field required to accomplish this is some ten billion times greater than anything Voyager could have generated, requiring a field amplifier literally as large as a planet. Sikaris has a mantle of tetrahedral quartz twenty kilometres thick, whose crystalline structure was used by the Sikarians as an amplifier.
Unfortunately, the Sikarians refused to allow Voyager to use their trajector and although members of the crew purchased the technology from a clandestine group on the planet the use of the mantle as an amplifier meant that it was impossible to use the device once they had departed. In addition, the trajector used antineutrinos as a catalyst for the space folding reaction, a technology incompatible with Federation technology.
Voyager was thus unable to use this technology to assist in its return home. No other species has been encountered which has this technology.