From Dragon Ball Encyclopedia, the ''Dragon Ball'' wiki
Rock-paper-scissors (じゃんけんぽん, Jan-ken-pon) is a game used from time to time as a means of deciding who gets to fight whom. Anyone playing rock-paper-scissors will say, "rock... paper... scissors..." and put their hand in a position so it resembles either a rock, piece of paper, or pair of scissors. Scissors beats paper, rock beats scissors, and paper beats rock. Four-way rock-paper-scissors can take quite a while.
Rock-paper-scissors is used at least one time in Dragon Ball and six times in Dragon Ball Z.
King Piccolo Saga
Dragon Ball Z
In the first episode, when Goku is telling everyone at the Kame House about Son Gohan, Gohan meets a Crab (which is much smaller than the giant one on Planet Namek) and they play rock-paper-scissors. Both of them keep picking scissors, as the crab can obviously only choose scissors, until Gohan finally picks rock. The crab, being a sore loser, sprays Gohan's face with bubbles.
When Goku falls off of the Serpentine Road into Hell and he asks two Ogres named Gozu and Mezu how to get out, they decide that if he can defeat them then they will show him the way out. It is implied that they are going to fight each other to determine who fights Goku first, but instead they play rock-paper-scissors. Gozu wins with scissors and challenges Goku to a wrestling match.
Rock-paper-scissors is played twice during the Namek Saga, though only once does it yield results. When the Ginyu Special Corps captured all seven Dragon Balls for Freeza in "Elite Fighters of the Universe... The Ginyu Force", Ginyu decided to fight Vegeta. He received complaints from Recoome, Butta, Jheese, and Ghurd because they all wanted to fight against Vegeta. He decided to take the Dragon Balls to Freeza and let Recoome, Butta, Jheese, and Ghurd play rock-paper-scissors to determine who fought Vegeta. Recoome won (he picked paper and the others all picked rock). According to Ghurd, Recoome always won, however, he may have just been sore that he had lost.
When Vegeta fought against Recoome in "No Refuge from Recoome", Butta and Jheese play rock-paper-scissors to determine who would fight aganst Kuririn and Gohan. They never finished, however, as they were interrupted when Vegeta unleashed a devastating attack against Recoome, and so they observed the fight from there until Goku knocked out Recoome and they were forced to step in.
When Babidi sent Puipui to fight against Goku, Vegeta, and Gohan in "King of the Demons", all three of them played rock-paper-scissors to determine who would fight against him. Vegeta won with scissors and easily defeated Puipui.
Kid Buu Saga
Goku and Vegeta lured Kid Boo to Planet Kaioshin in "Battle for the Universe Begins" for the final battle. Kibito Kai and Elder Kai were hopeful that they would do the Fusion Dance. Instead, Goku and Vegeta played rock-paper-scissors to decide who would fight against Kid Boo. Goku won with rock and fought against Kid Boo (this is the second time a game of rock-paper-scissors had concluded with Vegeta picking scissors).
- It is interesting to note that each side of the rock-paper-scissors triangle wins at least once (scissors wins twice).