Damn ol' brain. I just watched the performance of "The Weight" I referred to above; it was on "Evangeline"---with Emmylou Harris---that Richard plays drums and Levon mandolin. On "The Weight" Levon plays drums and sings the first verse, Mavis Staples the second, Pops the third, Rick the fourth, and all the final. A great version of this absolute classic of a song, better imo than anything John & Paul wrote. You may disagree ;-) .
I also failed to include The Byrds as bands having three lead singers (that is, when Gene Clark was a member). Great harmony vocals too, of course. Bassist Chris Hillman didn't sing while a Byrds member, but does in his post-Byrds career, both solo as a member of The Desert Rose Band.