I have seen this, back in 2002 if I recall correctly. In southern California. It was not stationary, but moved with the rotation of the earth, slightly SSW to NNE and seemed to be in the 1k to 2k foot range. It is NOT a chemtrail or other airplane residue. It is perfectly straight, seemed not to curve or conform to the curvature of the earth. I saw it over three nights, same time. No clue what it was/is but you are not alone in seeing it.

The only thing I can think of, is a faint view of the Milky Way, but you say the lines appeared in the foreground to the moon?

EDIT: It varies on how you see it, depending on time, location, light pollution and other factors, but here is a couple of pics:

I agree with the contrail shadow assessment put forth previously. I have seen contrail shadows many times.

Maybe something to do with the solar flares that hit yesterday? There were supposed to be auroras visible in some places, and maybe this caused an effect on contrails from a couple of jets? Just a guess, I don't know if that's possible.....just throwin it out there.....