Dragon Ball hasn't been especially kind to its version of Earth's moon. Besides being used to imprison Monster Carrot and the Rabbit Gang, it has been destroyed at least twice in the franchise -- once by Master Roshi in Dragon Ball, and later when Piccolo blew it up in Dragon Ball Z. The moon was also destroyed in Dr. Slump, which apparently exists in the same continuity as Dragon Ball. In fairness, Goku and friends have made at least two efforts to restore the moon with the Dragon Balls.
It's hard to say whether the moon was restored after being blown up in DBZ. In Dragon Ball, it's explicitly stated the moon was restored using the Dragon Balls. However, no such dialogue exists in DBZ, nor is any restoration depicted on screen. For all anyone knows, the moon is still, canonically, space debris.
Dragon Ball and Dragon Ball Z Both Feature the Moon Being Destroyed
The first time Dragon Ball's moon was blown up was during the 21st World Martial Arts Tournament. Master Roshi used a Kamehameha Wave to destroy it in order to undo Goku's Great Ape transformation. When Goku came back for the 23rd World Martial Arts Tournament, he explained that Kami allowed him to restore the moon in exchange for removing his tail permanently. This set the notion that the moon's restoration would have to be either stated or shown on screen.
The next time the moon was blown up came during Dragon Ball Z's early "Saiyan" saga. Like Master Roshi, Piccolo destroyed the moon with an energy beam in order to stop a rampaging Great Ape Gohan. However, unlike Dragon Ball, DBZ never had any scenes or dialogue explicitly indicating that the moon was brought back.
For the most part, the series seems to keep the moon's absence consistent. After all, it wasn't there when Vegeta wanted to use it to become a Great Ape himself. Besides that, most shots that would or could have the moon in them are noticeably vacant.
Despite the moon's consistent absence, there have been some shots of it since Piccolo blew it up. One was in Episode 118 of DBZ, "Frieza's Counterattack" where it can be seen as Frieza and King Cold are approaching Earth. The correlating Chapter 329 of the manga, "Where is Goku?" features a similar shot with a noticeably large, white speck over the Earth. However, assuming the moon is, in fact, still gone, this speck could just as easily be a distant star. If that's the case, then the anime interpreting it as a moon is a minor continuity error.
The Moon Has Reappeared Multiple Times After Piccolo Blew It Up
The moon is also visible in the 13th Dragon Ball Z movie, Wrath of the Dragon. Of course, since this is a film, the validity of this scene is questionable at best. The movies have a historically poor track record when it comes to keeping in line with the Dragon Ball continuity.
That said, the moon makes more blatant appearances in Dragon Ball Super. On the night before the Tournament of Power, it can be seen clearly in the sky. It's especially noticeable in Episode 95, "Most Heinous! Most Evil! Frieza's Rampage!" The moon is not only visible here, but it's enlarged for dramatic effect and even takes up most or all of the background in some shots. At the very least, this proves that the moon was brought back by the time of Super.
The moon's restoration by Super is further backed by the manga. During the "Galactic Patrol Prisoner" Saga, an extra page of Merus flying through space also shows the Rabbit Gang still on the moon. Of course, extra page continuity is about as trustworthy as movie continuity, so it should be taken with a grain of salt.
It's also worth noting that the moon is still completely absent by the time of Dragon Ball GT, which takes place after Super. This is especially clear when the Earth is viewed from Planet Plant, which was restored right in its orbit. Whether this continuity is still canon or not, it's meant to reflect the future of Dragon Ball after DBZ. The moon's absence here could imply that it was never truly restored, at least not at any point in Z.
In any case, it's safe to say that the moon is officially back in Earth's night sky. As for when and how it got there after Piccolo blew it up, that's still up for debate. Unless Dragon Ball GT is anything to go by, it should remain safely in orbit for the rest of Dragon Ball.