I'm going to disagree with Lurch - we're very sure the moon is geologically dead. We can quibble about "how dead is dead", but that's exactly what we are doing: quibbling.
There are moonquakes. Some are caused by tidal forces, some are caused by meteor impacts, and some are caused by thermal expansion and contraction of the lunar crust as it is alternately baked and frozen during the moon's long day and night. The type most like earthquakes are 28 events measured from 1969 to 1977 called "shallow moonquakes", which may be caused by (as Lurch said) by landslides in crater walls. These can register around a 5 on the Richter scale.
In any event, the rate of moonquakes is 28 over 8 years, or about 3.5 per year. In the last 365 days, the Earth saw 1637 earthquakes of that magnitude or higher. Put another way, looking at the 28 most energetic earthquakes in a similar period, the average energy is about 15,000 times larger on the earth.
So it's quite dead.