Why do interferometers only split a wave into two beams? I was recently reading about interferometers and was wondering why it isn't common practice to split a wave into three orthogonal beams? I know very little about the topic, but I thought if you were trying to characterize the directional aspect of gravitational waves it might be useful to be able to identify the waves components.
 A: This is a clever question.  In principle, it's not such a bad idea, and is surely practical for small-scale interferometers.  In practice, it would be hard to do for a gravitational-wave detector, and there are better proven solutions.
One of the problems with gravitational-wave detectors is that they are insensitive to certain polarizations.  (Google "gravitational-wave detector antenna pattern", and see the classic "peanut" shapes.)  If a single detector were to have three orthogonal beams, we could potentially combine their information to have three separate (but statistically correlated) "effective" detectors — one for each pair of directions among the three individual directions.  It would certainly be possible to have a first beam splitter reflect one third of the beam in one direction, then another beam splitter split the non-reflected part of the beam in two halves going the other directions.
This idea of having multiple partially overlapping interferometers is actually kind of like one of the original designs for the space-based version: LISA.  There, the three legs of the triangle could be split into three pairs of beams which would give you effectively three detectors.  (Last I heard, because the US pulled its funding from LISA, the mission had to "descope" and remove some of its components, which eliminated some of the equipment necessary for one of the pairs.)  Of course, LISA has just three satellites, which can only form a plane, but the core idea is similar.
Anyway, to answer your question of why we don't do it: There are several reasons, all of them practical.  First, LIGO/Virgo use big concrete and steel beam tubes to protect their laser beams in vacuum and prevent outside light from getting in.  You would need to tunnel 4km straight down (or tower 4km straight up) to make LIGO into a tri-axial detector.  Second, you would need to be able to hang a pair of mirrors such that they would have very low noise sensitivity to vertical motions, and be able to control them with excellent pointing stability.  Third, even after overcoming the practical hurdles, all this would buy you would be another couple detectors, but with highly correlated noise.  One reason the LIGO detectors were placed on separate sides of the US was to ensure small noise correlations.  Which ties into my last reason: Fourth, you could get the same advantage of having a detector in an orthogonal plane by just setting up a separate detector somewhere else in the world.  You wouldn't need to tunnel to change the plane of the interferometer; the curvature of the earth changes the plane for you.  Even though the two LIGO detectors are located on different sides of the continent, that's still not enough to substantially change the plane of the detectors.  This is part of why Virgo (in Italy) is important, and why LIGO Australia was such a good idea (but isn't going to happen because of politics), but LIGO India is still a good idea (and is going to happen because of politics).
