Fastest form of travel in the Star Trek universe

In the TNG episode "The Nth Degree," Barcley uses a subspace inversion to travel the enterprise a phenomenal distance, but it doesn't seem to be an actual mode of transportation. Maybe this is a one-off fastest form of travel. Are transwarp conduits the fastest consistent form of travel? So my question is what is the the fastest one-off or special form of travel, and what is the fastest standard or widely used form of travel? And what rate of speed are we talking about?

Answer

From the episode (see also here and here):

KIM: In principle, if you were ever to reach warp ten, you'd be travelling at infinite velocity.

NEELIX: Infinite velocity. Got it. So that means very fast.

PARIS: It means that you would occupy every point in the universe simultaneously. In theory, you could go any place in the wink of an eye. Time and distance would have no meaning.

In the episode, Tom Paris actually achieves this speed via a kind of "transwarp" technology developed for one of Voyager's shuttles, resulting in weird consequences for the character and one of the Star Trek franchise's most questionable products:

As you can't get faster than infinite velocity and occupying all points in space at once, this is the fastest one-off form of travel in Star Trek.

The fastest standard / wisely-used forms of travel that we have seen on-screen would be Borg transwarp conduits: while less than Warp 10, they greatly exceed Voyager's top speed of Warp 9.9 given what they allow Voyager to achieve in the final episode, "Endgame".

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