Unlike other planetary bodies away from Earth, the surface of the Moon is somewhat familiar to astronomers (the surface of the Moon is more easily observed with the naked eye than planets are with telescopes). The Moon does not have its own atmosphere as its gravity is too weak to maintain one. Its surface is marked by many meteor strikes, including among them Copernicus.
The DC Universe
The most notably variant aspect of the Moon as it stands in the DC Universe is that for the better part of the last decade, it housed the legendary Justice League Watchtower. Because of their presence there, the Moon experienced quite a few world-shattering events first hand. Throughout the Justice League Of America's tenure on the surface of the Moon, it has been pulled from orbit several times, nearly destroyed, and been hit by all manner of different weapons and entities.
However, with the advent of the JLA story arc "Crisis of Conscience", and the Infinite Crisis, the Watchtower has been destroyed by Superman Prime and Alex Luthor. Its towering ruins still stand to this day.
The Marvel Universe
The Moon of the Marvel Universe is notable in that it contains an area known as the Blue Area of the Moon. The Blue Area lies on the far side of the Moon in the Luther Crater. This portion of the Moon was artificially altered and given an Earth-like environment approximately 100,000 years ago as a result of a competition between two alien races, the Kree and the Cotati, to see who could create a better habitat. The Cotati won but the Kree, angry in defeat, attacked the Skrulls and initiated the Kree-Skrull War. Not only was the Moon the cause for this war that had a major impact on the Marvel Universe, it also became a home to various beings.
When the Fantastic Four made a trip to the Moon they found that it was where Uatu the Watcher lived and from where he watched all of humanity. Apart from Uatu, the Blue Area of the Moon was where the Inhumans made their home. The Inhumans moved their city (called Attilan) from Earth to the Blue Area after conflicts with Humans. The Blue Area also sustained the life of another alien race, but they died long before humans ever knew they existed and all that remains are ruins.
The other big thing that the Marvel Moon is known for is as the stage for the ending of the Dark Phoenix Saga. The Shi'ar, fearing the power of the Phoenix Force, bring the X-Men to the Moon where they must fight the Imperial Guard, with the winner deciding the fate of the Phoenix. The X-Men were losing and, after going crazy, Jean regained control and used a Kree weapon to kill herself and save everyone else. The X-Men were returned safely to Earth, and the Moon was left intact.
The Wildstorm Universe
The Wildstorm Moon doesn't have regular inhabitants like DC and Marvel, but a few things have happened there. In August of 1974, the Moon was moved from it's orbit by Mr. Majestic. It was the first test of his Planet Movers and first step in hiding the solar system from the Cosmic Negator. Not only was it moved though, it was moved in total secret. With machines set up all over the Earth and everyone watching the President resign on TV, the Moon was moved out of it's orbit. Another secret of the Wildstorm Moon is that it contains a Daemonite outpost. The base is small but has computers that hold massive amounts of information from all over the multiverse and also access to The Bleed.
The Bleed gates on the Moon provide a way to get to different dimensions and also let things from different dimensions into the Wildstorm one. Mr. Majestic got pulled into the gate by a Daemonite scout and ended up in the DC Universe. While he was gone from the Wildstorm Universe, Daemonites from another reality broke through and took over the planet Earth. Majestic eventually got back to his reality and, after traveling back in time to before the Daemonites got through, destroyed the base, so currently there is no Daemonite base on the moon. The last feature of the Moon is actually very similar to the DC Moon, if it exists. In a special issue of Planetary it was revealed that Planetary has a base on the Moon called the Watchtower. However, the issue (Terra Occulta) was crossover with the JLA and was an Elseworlds tale so there may not actually be a Planetary Watchtower.
In The Authority and tried to terraform the moon. It wasn't actually God, but that's what The Authority started calling the alien being that had sparked life on Earth. The whole process started right before the new millennium with giant, worm-like creatures landing on the Moon (and in Africa). In order to rid the Moon of its worm infestation (and also protect Earth), Apollo supercharged himself by taking a dip in the sun and then releasing all that energy onto the Moon, sterilizing it. After the Moon and Africa were cleared, the team took on God (a giant pyramid thing) and Jenny Sparks died killing it.
At the start of World's End, The High smashed one of his clones into the moon, destroying a large part of it.
In the Image Universe, the Noble Family has a base on the moon. Also, in one issue of Brit, a ship full of stoned aliens crash landed on the surface of the moon.
Other than the places on the Moon, the Moon itself has often been used as a place for superheroes to get away from the troubles of the world and think for a while. Most of the heroes that handle problems on a global scale and can fly and survive in space have at some point gone to the Moon for some peace and quiet. Superman types especially (for some reason) like the Sentry, Invincible and Superman himself frequently take time to relax in the comfort of the Moon.
There are also heroes that get their powers from the Moon. The most well known of these would be Moon Knight (or even Sailor Moon). Moon Knight gets his powers from Khonshu, the Egyptian moon god. There are also heroes who use the moon as a symbol even if they don't get powers from it. Midnighter, for example.