Covered Up: Some of their early hits, such as "Not Fade Away", "Time Is On My Side", and "It's All Over Now", are covers that have arguably displaced the originals in popularity.
"Harlem Shuffle", originally a Bob & Earl song from 1964 and covered by the Stones for Dirty Work in 1986, definitely fits here as well. It sounds so prototypically Stones that few even knew it was a cover before the days of the internet.
A slightly more complicated example would be "As Tears Go By" and "Wild Horses" (Sticky Fingers). Jagger and Richards wrote those songs, and the Stones' versions are definitive, but in both cases they were preceded by cover versions - they donated "Tears" to Marianne Faithfull before recording it themselves, while Gram Parsons convinced them to let his band The Flying Burrito Brothers cover the already-recorded "Wild Horses", and their version was released a year before the Stones' own.
Happened to one of their own songs. "I'm Free" is probably better remembered for the cover by the Soup Dragons. Thanks in part due to The World's End soundtrack.
Epic Riff: "Satisfaction", "The Last Time", "Get Off of My Cloud", "19th Nervous Breakdown", "Paint It, Black", "Jumpin' Jack Flash", "Brown Sugar", "Street Fightin' Man", "Sympathy for the Devil", "Gimme Shelter", "Rocks Off", "Tumblin' Dice", "All Down the Line", "Beast of Burden", "Shattered", "Start Me Up", "One Hit (To The Body)", "Mixed Emotions", "Can't You Hear Me Knockin", "Bitch"...
The "Satisfaction" epic riff came about when Keith Richards stumbled out of a hotel bed, recorded the riff, and promptly fell back asleep (the recording is eight seconds of riff and twenty minutes of snoring!). That's right kids, Keith came up with an epic riff in his sleep!
Keith Richards has said that if he were only allowed to play one riff for the rest of his life, he'd pick "Jumpin' Jack Flash".
The mid-70s albums are also considered this by some.
As is the post-Tattoo You albums to varying degrees.
That said, almost nobody wants to acknowledge Dirty Work.
Growing the Beard: Aftermath, the first album solely written by Jagger and Richards. Beggars Banquet is sometimes considered to have done this a second time, as it launched a four-album streak that is now widely regarded as one of the strongest series of releases in rock and roll history.
"New Rule: Airplane black boxes must be made out of Keith Richards. The Man who has done more drugs than Courtney Love, Robert Downey, Jr., and Rush Limbaugh combined recently fell out of a tree and crashed a jetski. And yet, that cigarette never fell from his lips. Something tells me the future of medical science isn't injecting stem cells, it's injecting heroin."
One suspected reason for his survival is that he has a very rare genetic mutation that renders him largely immune to the negative side effects of those drugs. Richards himself also notes in his autobiography that he always took special care to avoid increasing his dosage, which many drug users do not avoid doing and which is a sure way to increase one's risk of overdose (though it's still a risk even if one does exercise that much self-control, since the consistency of drugs can vary and thus it's not possible to be certain how much, or for that matter even what, one is taking). To be fair, his genetic mutation also may be one reason he was even able to exercise that much self-control in the first place.
Memetic Mutation: Thanks to his sightings in the 2010 FIFA World Cup, Mick Jagger has now become a harbinger of bad luck for any team he decides to cheer on, like the English Team, for example. Even though he's not to blame that that English team had such a low technical level.
Seasonal Rot: Some fans say they haven't been good for a long time. The most frequently cited "last good/great album" is Tattoo You, while many listeners and critics cite Some Girls as the band's last genuine masterpiece.
We're Still Relevant, Dammit!: Justified with 1978's Some Girls. Music critics had written off the Rolling Stones as outdated with the emergence of Punk Rock and disco, but the Stones thought otherwise with Some Girls - which proved popular with critics and listeners alike. It helped that Mick Jagger was a keen follower of the punk and disco scenes in New York and London. The fact a lot of punks grew up listening to the Stones doesn't hurt either.