In Star Trek, space is two-dimensional.
...Or perhaps sub
That is, there are a total of three cases where the ship is in danger of being "hit by a 2D wavefront" in Star Trek. The first is in the second pilot episode, "Where No Man" (and its repeats in "Is There In Truth" and "By Any Other Name"); the second is in TNG "New Ground"; and the third is in the movie ST6: The Undiscovered Country. In the latter two cases, the flat wavefront is supposedly moving at warp speed - in "New Ground", explicitly so although in ST6, only implicitly reaching out to a Federation starship from deep within Klingon space. The weird properties of subspace might well dictate the "channel" within which the wave propagates, leaking back into the geometry of real space in the form of a flat ring. Similar things might hold together the Galactic Barrier of TOS in a flat if seemingly static shape.
Now, if a natural phenomenon finds it advantageous to move in a specific part of subspace (and its realspace counterpart), one might assume a starship does, too. Which means Sulu's ship in ST6 would justly be caught in the "sweet zone" while preparing to get underway for the long warp home, and would have no time to go up or down. OTOH, Picard's ship in "New Ground" needs to hang around the wavefront for mission-specific reasons, which don't really need explaining.
Which basically only leaves the TOS barrier. Why not go under or over? Well, Kirk says his mission is to ensure that future missions in this direction will be safe - so he deliberately sails into danger, to check it out. The Barrier doesn't appear to come as a surprise to him or his crew, only its properties do. Quite possibly, Kirk was given the specific orders to go and muck around inside the Barrier to see if it's safe - and not the orders to go past
it and find wonderful things beyond, as nowhere in the episode is there mention of a desire to do the latter. That's for other starships.
Really, the 2D vs 3D problem doesn't appear particularly severe in Star Trek. Most of its "space waves", "ion storms" and whatnot are delightfully 3D anyway. And as for more physical phenomena... Well, in TNG "Final Mission", there's a realistically 2D if unrealistically dense asteroid belt between the ship's starting point and her destination, and Riker opts to fly through rather than above or below. In terms of today's celestial mechanics, that makes sense: trying to change the plane of one's orbit would be hard work. Sure, Trek engines are better, but the episode does establish that the Enterprise
is straining with her towing task already, and is in a great hurry. So going through the asteroids is probably a good idea after all, even if the superdense belt poses a collision risk.
What else? Ah, the minefields in DS9 "Sons of Mogh". Supposedly, they are in the same plane as the Bajoran system's planets, rather than in a 3D surround pattern. Of course, the purpose of a minefield never is to surround: it is to unpredictably ambush, so that the opponent never knows where and whether
the mines are. A thousand mines at the plane of the ecliptic is much better than a trillion mines evenly spread. Provided that the ecliptic is a shipping route, that is.