Lennon - Dylan 
John Lennon and Bob Dylan were friends since their earliest days of stardom; but 13 years later both were in different headspaces; Lennon had retired from music to be a stay-at-home dad while Dylan was a born-again Christian cutting religiously themed material. One example was “Serve Somebody,” which Lennon was reportedly incensed by; objecting to its “mediocre” musicianship, “pathetic” vocals and “embarrassing” words.
Harrison - McCartney
Disagreement between Paul McCartney and George Harrison boiled over during the filmed sessions for the Beatles’ album Let It Be, when, while rehearing the song “Two Of Us,” Harrison became sufficiently fed up with McCartney’s instruction and left the band. The band completed two more albums before breaking up, after which Harrison released his first proper solo album, All Things Must Pass, a three-disc set -- he later complained that McCartney’s emphasis of his own songs led the band to disregard Harrison’s, which is why the album was so long. Harrison’s disdain for McCartney lasted for years; in 1984, when McCartney issued Give My Regards to Broad Street, an album of re-recordings of his old songs, Harrison snipped in a television interview that he’d lost his creativity.
McCartney - Yoko Ono
To say Paul McCartney and Yoko Ono got off on the wrong foot would be putting it mildly. According to John Lennon, when he and Ono were staying with McCartney when they first started dating, McCartney left them a note that said “You and your Jap tart think you’re hot shit.” McCartney has complained that she’d even sit on his amp and he’d have to ask her to move to adjust the volume (in fairness, Ono later claimed that if she hadn’t gone with him, Lennon wouldn’t have showed up at all).
Allen Klein - Beatles and Stones
When the Beatles and the Rolling Stones first shot to fame in the early ‘60s, the press depicted them as rivals. This was fabricated; in reality, the two bands were friendly. The Beatles wrote a song for the Stones, members of both bands played at each other’s studio sessions and they even coordinated their album releases to avoid overlap. Who the Beatles and the Stones really had beef with was manager Allen Klein. When he took over management of the Beatles’ Apple Records, he quickly fired many employees and alienated the rest. McCartney’s disagreement with the other three Beatles about his involvement in the band was a key factor in its breakup. In managing the Stones, meanwhile, Klein tricked the band into signing over the rights to most of its songs recorded prior to 1971.
Mike Love vs. The Beatles, The Rolling Stones
The Beach Boys are well known for hating and constantly suing each other, but when the band was inducted into the Rock and Roll Hall of Fame in 1988, singer Mike Love turned the invective outward, taking the occasion to put what he perceived as his competition on blast. Speaking last (tellingly, his band mates all left the stage), Love first called out Paul McCartney, “who couldn’t be here tonight because he’s in a lawsuit with Ringo and Yoko,” terming such “interstistine [sic] squabbles” a “bummer,” adding that “the mop-tops” couldn’t match the Beach Boys’ 180 performances a year. He then moved on to the Rolling Stones, stating “I’d like to see Mick Jagger get out on the stage and do ‘I Get Around’ versus ‘Jumping Jack Flash’ any day,” later adding, “He’s always been chickenshit to get onstage with the Beach Boys.”
Simon - Garfunkel 
The two folk musicians recorded their last album, 1970’s highly successful Bridge Over Troubled Water, at a time when their personal relationship was rapidly deteriorating. The two couldn’t decide on the 12th and final song on the album and refused to record each others’ choices, so the album was released with only 11 tracks. Art made the mature decision to put the past behind him and publicly praise Paul for his songwriting ability. Paul responded by saying, “Arthur and I agree about almost nothing, but it’s true, I have enriched his life quite a bit.”
