Right so I've actually been tracking this for about three years now and there does seem to be something in it, though I'll admit the sample size from my own notes is too small to be conclusive. The cases I've logged from witness accounts in the Lincolnshire area cluster noticeably around new moon and the three days either side of it. Not full moon like people always assume.
What I find more interesting is whether the moon phase correlation is causal or just correlational. Darker skies around new moon means more people are outside stargazing or doing rural stuff, so naturally you'd get more sightings reported. That could entirely explain the pattern without needing any lunar mechanism.
Has anyone actually done a proper statistical breakdown against a decent sized database like MUFON or NUFORC? I had a go at it a couple of years back but teh data categorisation on those reports is inconsistent enough to make it a nightmare. Would genuinely like to see someone do this properly rather than just anecdote stacking.