I don't really do "bands" as such, but in terms of recording artists in general, my favorites list would be something like this (in no order):
The Beatles
Prince (1979 to 1996)
The Stone Roses
Stevie Wonder (1968-ish to 1982)
Bob Dylan (1961 to 1983)
Joni Mitchell (to 1994 or so)
U2 (1980 to 1993)
The Byrds (1965 to 1968)
The Police + Sting (to 1993)
The Wailers / Bob Marley & The Wailers
John Coltrane (late 1950s to 1964)
The Doors (to 1971)
Joan Baez
Simon & Garfunkel / Paul Simon (to 1990)
Sinéad O'Connor (she's a complete nutter, but all her recordings are great... except maybe the reggae one)
Louis Armstrong
Billie Holiday (to 1948 or 1949)
Counting Crows (studio, not live! to maybe 2004)
Robert Johnson (yes, based on only two recording sessions)
Indigo Girls
I also like very much like early Elvis, Gordon Lightfoot, George Harrison, John Lennon, Björk, Gladys Knight & the Pips, Johnny Cash, The La's, Yazoo, The Grapes of Wrath, and R.E.M.
Artists whose recordings I've started to really enjoy recently include Clannad. There's a lot of stuff there.
Probably my favorite contemporary singer is Michael Kiwanuka... especially if he ever comes out with another album.
The one I've listened to the most in recent months is Prince. Once you get into the B-sides, bootlegs, and outtakes, it's an endless treasure trove... that is, if Prince doesn't sue you and confiscate your computer.
I'm not sure where U2's peak period of recordings would "rank" in all this. I am sure it's not important.