Banned in China: The series was banned in Saudi Arabia, Thailand and United Arab Emirates.
Dueling Games: With the Driver series initially, thanks to some not-so-friendly jabs that III took towards that series. In the IV era, the Saints Row series became its main rival, with ads for Saints Row 2 mocking IV's perceived stonefaced seriousness.
Fan Nickname: 'Flipstalker' for the Landstalker in the 3D era games for reasons that are apparent for anyone who has tried doing sharp turns with it.
Hey, It's That Guy!/Hey, It's That Voice!: Oh. So. Much. Even in IV, which had unknown actors for the major parts, they still cast celebrities to do DJ work for the radio stations (the big ones being Juliette Lewis for the indie rock station, Iggy Pop on the classic rock station, and Daddy Yankee on the Latin station). San Andreas has Axl Rose as the DJ for K-DST.
Oh God, San Andreas had a ton of these. Perhaps the most prolific were the rumors that Bigfoot (or a Sasquatch, or a Yeti) roamed the forests, and that there were "ghost cars" that drove around in the countryside. The former turned out to have been faked with judicious use of Photoshop, while the latter was a result of abandoned, wrecked cars spawning atop hills and then rolling down them (quick note: when cars spawn, they don't have their parking brake on). Other rumors included zombies (mostly the result of a building in San Fierro that housed a company called "Zombocorp"), UFOs, the Loch Ness Monster, Jaws, and a masked slasher who ran around the woods with a chainsaw.
IV also had a few. There were rumors that Lola, a prostitute who appeared on some of the promo art, could be found in-game, that a creature called Ratman could be found lurking the subways, and that an abandoned factory is haunted.