This article is from the Star Trek Tech FAQ, by Joshua Bell inexorabletash@hotmail.com with numerous contributions by others.
"Where No One Has Gone Before" [TNG]
The Traveler propels the Enterprise to the Andromeda Galaxy,
using more advanced warp equations.
"The Nth Degree" [TNG]
The Cytherians bring the Enterprise to the center of the galaxy
via graviton mediated subspace folding.
"New Ground" [TNG]
Soliton waves are used to move a ship at warp speeds; "Warp
without warp drive."
"Contagion" [TNG], "To The Death" [DS9]
The Iconians had portals to other places in the galaxy.
"The Price" [TNG], ST:TMP, and DS9
Wormholes, stable and unstable, connect remote parts of the
galaxy. In Star Trek: The Motion Picture an unbalanced warp
field causes an artificial wormhole to appear around the
Enterprise.
"Interface" [TNG]
Geordi proposes an artificially created subspace "funnel"
connecting points 300 light years apart.
"Caretaker" [VOY]
The Array is able to bring ships across the galaxy with some
form of displacement wave, which looks very similar to the
shockwave seen in Star Trek VI.
"Prime Factors" [VOY]
Space-folding technology can be used from the Sikarian planet
to transport people up to 40,000 light years in an instant.
"Explorers" [DS9]
Sisko builds a solar sailing ship. When out sailing with Jake,
a tachyon stream "catches" (ugh!) the sails and the ship is
accelerated to the FTL speed of the tachyons. [Michael Welch]
"Vis a Vis" [VOY]
Co-axial warp drive. Apparently Federation scientists have been
working on this for years. [D. Jacobs]
In several episodes ("When the Bough Breaks" [TNG], "Tin Man" [TNG],
"Q-Who" [TNG], "The Gift" [VOY]) some unknown technology or
non-corporeal entity throws a starship tens, hundreds or thousands of
light-years in mere seconds. In those cases there is rarely associated
technobabble to hint at the underlying mechanism.
(I'm missing lots, please let me know.)
 
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