Ray Charles did a, well, Ray Charles version of "Swanee River."   Very r&b-soul-bluesy.

UB40 did a reggae version of "Can't Help Falling In Love With You" (in which he unforgiveably changed the words "Like a river flows surely to the sea" to "Like a river flows gently to the sea" thus totally missing the point of that lyric line but I digress).

"Summertime" from Porgy & Bess was an operatic aria but has been performed and recorded in just about every genre and every era's style.

Paul Simon's "Bridge Over Troubled Water" was a slow ballad when he and Art Garfunkel sang it. Both Gladys Knight and Clay Aiken recorded it as a powerful, more rhythmic, more up-tempo Gospel-style song. Johnny Cash's version sounds little like the original. It sounds like, well, it sounds like a Johnny Cash song.   Many other versions are out there too.