Several appearances include: The St. Anne scheme, capturing Pokemon that Mewtwo weakened, and doing construction work around Clarity Lake.
It seems that all Team Rocket members begin their criminal careers in this position, and then work their way into the higher ranks. Most of these Rocket Grunts are male, but there are a few female members as well.
In the GameBoy games, the Rocket Grunts carry out most of the actual work in Giovanni's organization. Some of these schemes include stealing fossils from Mt. Moon; causing trouble at Silph Co., Lavender Tower, and Cinnabar Lab; cutting off Slowpoketails and selling them for high prices; forcing useless Magikarps to evolve into powerful Gyarados; and occupying the Goldenrod Radio Tower to transmit their own messages. While your trainer causes TR to disband in R/B/Y with the defeat of Giovanni, the minions and the Executives continue their criminal activities, waiting for their Boss to return.
As long as Team Rocket continues to operate, there will always be a need for underlings to do the dirty work. Look for lots more Rocket minions in future Pokemon games and episodes.