To what extent did both rock and roll fans and the Beats offer a serious challenge to American society in the years 1955-63?

The rock-and-roll fans who gravitated toward Elvis Presley were white. So were most of the Beats. However, the teenagers who consumed rock were otherwise mainstream. They were "bobbysoxers"—the supposedly clean and wholesome products of upstanding, middle-class white families. Their interest in rock, a sexually provocative music associated with the black people who had invented it, presented a challenge to notions of propriety among white youth. There were fears that rock-and-roll (named after the movements of love-making) would encourage sexual activity among young people. There were also concerns that an affinity for this music would encourage integration. It is true that the rise of rock-and-roll coincided with the Civil Rights Movement. The Beatles, for example, publicly refused to play before segregated audiences.
The Beats were pejoratively referred to as "beatniks." Beat culture arose in North Beach near San Francisco in the postwar period. "Beatniks" were young men and women who wore all black. They wore berets, too, to signal their alignment with French existentialism. They wore sunglasses at night, spoke in the rhythm of the jazz they loved, they associated with black people, and—most shockingly—they smoked marijuana. The image of the beatnik was the antithesis of the well-scrubbed bobbysoxer. However, the bobbysoxer's love of rock-and-roll created the fear that those wholesome teenagers would veer more closely to the counterculture embraced by the beatniks.
The image of the beatnik was designed to make the Beats look foolish, thereby diminishing their ideas. The actual Beats, represented by the triumvirate of Jack Kerouac, Allen Ginsburg, and William Burroughs, were white men who had eschewed the values that most white men had embraced. They did not desire careers or wives (Ginsburg and Burroughs were gay, and Kerouac was probably bisexual) or houses in the suburbs. They traveled frequently and lived where they wanted. They did drugs. They disliked the ways in which white society limited sexuality and hindered their interactions with people who were not white.
Both rock-and-roll fans and the Beats wanted something new—a new sound or a new mode of living, alternatives to what mainstream society valued as normal and good. The "serious challenge" was simply their courage to be different in an era of extreme conformity.