The main problem is that the Moon has many mascons, not a uniform sphere. Close orbits around it tend to get perturbed. They keep the same semi-major axis but become more and more elliptical until they hit the Moon. Early satellites sometimes hit the moon within weeks, while others lasted for months or longer, and it took them a while to figure out why. Contrary to what some articles say on the web, its Hill sphere is easily large enough to permit moonlets to orbit the moon, the problem is, the mascons.
from the NASA page: Bizarre Lunar Orbits:
"There are actually a number of 'frozen orbits' where a spacecraft can stay in a low lunar orbit indefinitely. They occur at four inclinations: 27º, 50º, 76º, and 86º"
That's especially so for surface skimming orbits, or ones that get quite clos. For more distant orbits - the most stable orbits are retrograde. A retrograde orbit around the moon, looked at another way is an elliptical prograde orbit around the Earth, with the same orbital period as the Moon. When closer to the Earth then it orbits it more quickly than the moon and when further away, then it orbits more slowly than the moon, so combined effect is that when seen from the moon it orbits it in a retrograde direction.
You can get retrograde orbits both inside and outside of the lunar L1 and L2 positions - i.e. either inside both of them, or outside both of them, relative to the moon.
These orbits are stable over time periods of about a century or so. But not necessarily over longer time periods. With the NASA asteroid capture mission, one of the proposals is to return it to a lunar retrograde orbit.
So far though there are no known natural objects in any lunar orbits.
However we wouldn't be able to see really small objects of just a few meters in diameter if they were orbiting the moon. So that leads to a natural question - could it have temporary satellites that we don't know about, which eventually either hit it, or escape to interplanetary space via the Sun Earth L2 (by coincidence the Sun Earth L2 is accessible to either of the lunar L1 or L2 points by a low energy transfer with almost no delta v).
You do get temporary extra moons of the Earth, which orbit it for a few months and then leave it again. Such as 2006 RH120, 2-3 meters across which was a satellite of the Earth for nearly a year: September 2006 to June 2007. It's estimated that at any particular time there is probably at least one temporary moon of the Earth one meter in diameter, and perhaps 700 or so tiny moons at least 10 cm in diameter.
Also there may be dust in the L4 and L5 positions in the Kordylewski cloud if it exists - temporary because those are unstable in the Earth Moon system.
So given that population of tiny moonlets of the Earth, and these L4 and L5 transient continually replenished and dissipating clouds (if they exist), could there be some moonlets at least 10 cm in diameter in 28 day elliptical orbits around Earth that make them temporary retrograde moonlets of the moon?
But I don't know of any similar result for the Moon.
Indeed there are, so far, no known moons anywhere in the solar system that themselves have moons or rings. At one point Rhea, moon of Saturn was thought to possibly have rings. It still remains a possibility, though the evidence is indirect and inconclusive and a search for a ring hasn't turned up anything. A ring is basically a large number of very small moonlets.
We don't yet know of any moon in our solar system with moonlets or rings orbiting it, or indeed, don't know of any "double moon" although double asteroids are quite common.The nearest to this are Saturn's two co-orbital moons Epimetheus and Janus.
You might be interested in my article on this topic, which I wrote as New Horizons was approaching Pluto hence the title - to make it topical - but of course New Horizons didn't find any new moons.
Can Moons Have Moonlets? Or Rings? Moonlets Of Pluto's Moons?