The 1972 North American Tour promoting Exile on Main St.. The Texas shows captured on film in "Ladies and Gentlemen, the Rolling Stones".
During one show in 1981, shortly after John Lennon's murder, when rockstars were understandably wary of Loony Fans, the Stones were in the middle of "Satisfaction" when Keith spotted a lone fan climbing on stage and making a dash for Mick. Keith calmly turned down the volume on his guitar, hit the guy over the head with it, strapped his guitar back on, turned up the volume and kept playing without missing a beat. Then he went down to the police station and bailed the guy out.
They announced their 1975 North American tour by calling a "press conference" at a hotel in Midtown Manhattan, then pulling up to the front entrance on the back of a flatbed truck loaded with amps and instruments and performing "Brown Sugar" on the spot.