What event happens once every month

There are many many of those.  Here are a few:

-- I get paid on my job.

-- Dates numbered from 1 to 28 .

-- I receive a social security check.

-- I receive a small pension payment from BP.

-- I make payments on my home mortgage and car insurance.

-- If there is any balance due on my credit card, I pay that too.

Note:  Any answer that says anything about the Moon is incorrect.  The Moon's period of orbital revolution is 27.32 days, and the period of its full set of phases is 29.53 days, so there's nothing about the moon that always happens once and only once in a month.  For example, there can be either one OR TWO New Moons or Full Moons in the same month, and although it's rare, mathematically difficult, and unusual, you can have a February with NO new moon in it, or a February with NO full moon in it.  

The Moon's sidereal orbital period

Explanation: The period of the moon is almost equally to a month, this means that you usually would see a new moon once per month, now, this is not exactly true, because the period of the moon is not equal, so there may be months where there are two new moons for example (but some of the other phases of the moon only happens one time) Now if we consider the full sideral period, this would start or end only once per month