Would it be scientifically useful to put a LIGO or VIRGO on the moon?
Scientifically at a first analysis it would make only a small difference. The reason is that there are other sources of noise besides the terrestrial and man made noise, that together are about the same amount of noise.
It's easy to see it in the following summary of the LIGO amplitude spectral density sensitivity of about $10^{-23}$ per sqrt(Hz). http://ligo.org/science/Publication-O1Noise/flyer.pdf See Figure 3 specifically, the seismic and Netwon noise is one of the the plotted curves, it seems the dark grey one, but in any case of approximately the same order of magnitude as more than 10 other noise sources that have to do with the measuring apparatus, and not external noise limited. The red curve is the measured noise, the purple one is the expected noise (probably RSS of the others). It is clear that it is not mainly external noise limited.
So if you placed this Ligo on the Moon, and were able to control all the Moon Related problems as well as on Earth, the sensitivity would not change, for the most part.
Now, it is possible that many of the non-seismic noises were designed that way because it was useless to do better on those and have it all be seismic dominated. You'd have to read detailed papers and designs of the apparatus. It took a long time to design it and build it, this second generation took 3-5 years, the first more than 10.
LISA will definitely do much better. It is much longer so a strain would cause a displacement larger by the ratio of the lengths (see below, this is true for longer wavelengths). LISA, the original space based interferometer was to have 5 million km legs (compare with a few Kms for LIGO), and distances and measurements would be done by an advanced design as well. The latest LISA is proposed to be 2 detectors instead of 3, so you loose a little, but still much better than LIGO because of the much longer lengths. You can see the NASA website or Google it. See the NASA site at https://lisa.nasa.gov. Arm lengths may also be a little less.
There is a review, but I can't locate it, that shows the sensitivities and spectral range, as well as what kind of object emit those frequencies. It might be a Living Review of Relativity, but can't confirm it.
EDIT/ADDITION ON LISA SENSITIVITY AND INTERFEROMETER MEASURE IN SPACE
First, changed the name of the interferometer 'legs' to 'arms' to make it consistent with standard usage on the topic.
More importantly added more on LISA.
I am adding this response to a very good question in the comment by @robert bristow-johnson below. He asked how does the interferometer work in space since the arm lengths do not appear to be rigidly fixed. In fact there are a few things that are done to try to measure the path length changes due to ONLY the gravitational forces (or changes in spacetime curvature, equivalently). The first part is that a drag-free satellite is used so that non-gravitational effects (like solar wind and light pressure ) are eliminated (hugely reduced). A drag-free satellite uses the satellite itself as a container, but lets the detector test masses float inside the satellite and follow spacetime geodesics, i.e. freely floating trajectories. Sensors and small jets keep the container, the satellite, centered around the test masses. See the descriptions of such things in https://en.wikipedia.org/wiki/Zero-drag_satellite.
There's more that they have to do, besides as you'd think besides making sure the sensors and jets don't affect the test masses too much. The arms are not rigidly locked, and they have to keep track and try to offset long term effects (such as changes due to planetary movement), which would be in essence pseudo-static and not gravitational waves. They measure the arms lengths it with lasers, actually keeping track of how many millions of wavelengths are changing constinuously, over short and longer time periods. They separate those changes in the frequency domain and offset and filter out the long period changes while using the rest to look for the gravitational waves.
There's more to the whole story of detecting gravitational waves in space. Since the arms are not rigidly locked they can't use Fabry-Perot cavity type interferometers. The end result is their sensitivity is about 3 orders of magnitude less in strain spectral density, so $10^{-20}$ instead of the number above. On the other hand the arm lengths are about a million times larger, way more than offsetting that for wavelengths larger than the arms length (for smaller wavelengths it's worse). See the description of LISA and some of these issues, as well as one of the latest parameters in the Wikipedia article at https://en.wikipedia.org/wiki/Laser_Interferometer_Space_Antenna
Bob Bee's answer is a good first cut answer addressing signal to noise issues. I also find it surprising. I make some notes on my reading of Bob Bee's answer and the sources he cites in the footnote.
But signal to noise is not the only issue at play here and there may be some merit in putting detectors on the Moon. The triangulation ability of the two current LIGO stations is limited, but an array of telescopes will be able to fully determine the direction whence gravitational wave signals are coming. The bigger the array, the better the angular resolution. For the measurement of sources which we have no corresponding light emissions for (such as neutron star collisions), say where the events do not emit light or where the light is blocked by interstellar dust, we will in the future turn to images built up from gravitational waves to help probe inaccessible-to-light / radio astronomy regions of the universe.
If we put part of our LIGO array on the Moon, the possibility of very long baseline interferometry arises. Of course, the same could be done for a large freefalling array such as LISA, but there could be some engineering / economic advantages for a Moon based member of such an array.
More on Signal to Noise
There are also some very good summary noise breakdown curves in the Einstein Telescope design documents. If I'm reading the SNR curves right (i.e. those such as Fig. 3 in the LIGO Summary cited by Bob Bee, it would seem that LISA won't improve the detection threshold much for the kinds of events that have already been seen (i.e. ≥100Hz) as the noise becomes quantum dominated for these frequencies, but there is about an order and a half of magnitude improvement possible for lower frequency events. However, it's not clear whether the "quantum noise" curve is simply the $\sqrt{N}$ Poisson photon arrival noise or whether they've already taken possible squeezed light state improvement into account. The use of squeezed light to improve phase resolution at the expense of amplitude is planned for the Einstein telescope.