This might be fun. Or it might be useless. (Many things are both, I suppose.)
I'm talking purely about originals, of course. Paul was usually the one to try singing like someone he admired, just to escape the limitations of his own mannerisms and lend a certain feel to a song. He tried to sing, strangely enough, like Marianne Faithfull in "Here, There and Everywhere." Hence the nearly falsetto voice.
"Lady Madonna" was an attempt to be Fats Domino (Paul was very gratified when Fats later recorded it), and the obvious Little Richard homage is "I'm Down." I can't figure out who Paul's trying to sound like on "She's a Woman," but he sure sounds goofy!
"I'll Follow the Sun" is sung overtly in the style of Gerry Marsden. One wonders if the song was originally resurrected as an offering to the Pacemakers, but was used on Beatles For Sale because of the shortage of originals at the moment. On Anthology 3, it's apparent (to my ear, anyway) that "Oh! Darling" was originally done in the Smokey Robinson style. Paul almost sounds like Kermit the Frog.
The initial version of "No Reply" on Anthology 1 is sung like Tommy Quickly, the guy for whom the song was written in the first place (before John decided he liked it enough to keep it). Note the funny, un-John-like way the title's sung at the beginning: "No reply-hy-hy-hy." It almost sounds like Billy J. Kramer, oddly.
The only other instance in which I can think of John trying to change his elocution, tone, etc. to sound like someone else doesn't really involve a mild imitation at all; he's just deliberately adopting a very thick Scouse accent during "Polythene Pam."