"You'll Be Mine", released on Anthology 1, is clearly not a song that's not meant to be taken seriously. John and Paul sing in ridiculous operatic, falsetto, and faux-bass voices, and the songs ends in a burst of laughter.
Nevertheless, it does suggest some of what The Beatles were listening to- namely, doo-wop. Doo-wop is probably the oldest genre to survive through the '50s, having beginnings in the early '40s- in fact, the group that could be considered the fathers of the genre, the Ink Spots, are who "You'll Be Mine" is often compared to.
Doo-wop is a somewhat loose term, having not been coined until the early '60s when it had already been around for a while and making lots of hit records. Stereotypically, doo-wop groups are African-American vocal harmony groups whose consist of a lead tenor being backed by a baritone, countertenor, and of course, a bass vocalist, making frequent use of meaningless hum-like syllables like "shooby-doo-wah" and "ba-dum" and "wah-ooh". It has roots in gospel and blues with pop thrown in for good measure, probably having its genesis in four or five guys singing at a street corner, to romanticize the genre's image a bit. Of course, doo-wop does not necessarily have to have a twelve-bar form or consist of I-vi-IV-V chords, because doo-wop is a surprisingly flexible genre from what I've observed, having lots of moods, tempos, and chord structures available.
Of course, I don't really know how easily The Beatles could've heard the Ink Spots' records, since they didn't really exist by the late '50s, but I guess anything's possible. Nevertheless, its a common facet of doo-wop music to have a middle where the bass singer speaks a monologue, which is what I'm guessing to be John sends up when speaking of his love's "National Health eyeball". I don't see who else it could be, with such a nonsensical phrase.
So there you go- The Beatles were flavored with a sprinkling of doo-wop! Why else would they use "shooby-doo-wop" on "Revolution 1"?
By the way, since there isn't really any other place to mention it, there's an incomplete fragment of an unknown song from the 1960 home recordings. It's a curious, somewhat jazz-ish number from what can be heard, with an unusual ascending chord structure utilizing diminished chords that makes me doubt that it's one of their compositions, with a "ba ba ba" vocal by Paul. Then again, I did read an interview once where one of them said that they would learn a new chord and write a song around it, so who knows? It's just that I don't believe I've heard them use diminished chords any anywhere else, besides "Because" and "Old Brown Shoe". Anybody got some examples?
UPDATE: I now recall that "Like Dreamers Do" utilizes a diminished chord, so I guess it's not such a stretch after all.