"The Judgment of Paris" has been a popular story for a super long time. It was so popular that Homer doesn't even bother to go into much detail about it in the Iliad, his epic account of the Trojan War. He was probably like, "Oh that one about the golden apple and the beauty contest? Everybody knows that one. Let's get into the nitty-gritty of the wrath of Achilles. That'll be awesome." (Yep, we're pretty positive that's an exact quote from Homer.)
Some say that Homer actually did write a full version of the story in the epic poem, Cypria, which was a prequel to the Iliad included in what's called the Epic Cycle. However, only fragments of this epic remain, and most people highly doubt that Homer wrote it, saying that either a dude named Stasinus or guy named Hegesias wrote it instead.
Today we can find pretty detailed versions of the story in the writings of later poets like Ovid, who included the story in his Heroides and Amours. Hyginus also gives us most of the deets in his Fabulae. There's also a version in book by Apuleius in which the narrator describes watching a play version of the tale. Oh, we forgot to mention that the name of Apuleius's book is The Golden Ass. (No, he means a donkey.) The myth also became super popular with the painters of the Renaissance and beyond. Big names like Rubens, Regnault, and Dali all took a swing at depicting those lovely goddesses.