A classic anecdote, particularly among such people as police and paramedics, is that (for example) the real nutters come out when there's a full moon. They don't usually go so far as to say that with every full moon comes a night with nothing but nutters, but do seem to suggest that there is a significant trend. If you want a good selection of anecdotes on this score, head over to the Yahoo! Answers page, "Full moon - does it bother you?" (be prepared for some industrial-strength ignorance).
The full moon has also been popularly associated with all sorts of things, like rises in suicide rates, crime rates, accident & emergency admission rates, alcoholism, natural disasters and many many more. Here's the important factoid for you to remember: there is no evidence to support these hypotheses. And yes, studies have been done.
What this is is a classic example of confirmation bias - one only notices the events which confirm the hypothesis that one has (consciously or otherwise) adopted. I won't spend a lot of time debunking all of this - it's a "Blink" post after all - so I'll just point you at a very useful SkepDic entry: "Full moon and lunar effects". There's a substantial reading list at the foot of that article if you want to look further into this subject.