Over the past two days, I listened to two discs of a compilation album of Gene Vincent. It's his first two albums with some bonus tracks. In any case, some of the backing vocals during "Wear My Ring" sounded pretty familiar, so I checked some things and discovered that the backing vocals in the Beatles' "In Spite of All the Danger" are comprised of the same four notes.
In "Wear My Ring," after the line "To remember when December comes in May," the backing vocals go from B to E to D# to D. The backing vocals in "In Spite of All the Danger" use those same four notes. There's a slight difference in that the note values in "Wear My Ring" are all the same, where "In Spite of All the Danger" stays on the E a bit longer.
The chronology and influence match up too. According to the liner notes of Anthology 1, the pre-Beatle Quarry Men recorded "In Spite of All the Danger" in the spring or summer of 1958. Gene Vincent's "Wear My Ring" was recorded in 1957. Anthology 1 also contains the Beatles' version of "Ain't She Sweet," and the liner notes mention that "the Beatles usually performed this in the more mellow style of Gene Vincent and his Blue Caps' influential 1956 version." So the Beatles held Gene Vincent in some regard, and it's chronologically possible that they based the backing vocals in "In Spite of All the Danger" on those in Vincent's "Wear My Ring." (As a sidenote: the song was written by Bobby Darin and Don Kirshner, but that phrase doesn't appear in the backing vocals in Darin's version, which was actually released after Vincent's version, in 1958.)
I've read that "In Spite of All the Danger" bears some resemblance to Elvis' "Trying to Get to You," so this resemblance to a Gene Vincent song has some precedent. It's like "In Spite of All the Danger" is an amalgam of the Beatles' early influences.