Jamaican singer Shaggy of “It wasn’t me” and “Boombastic” fame sounds nothing like he does in his songs, he actually speaks without an accent. He once told a story about how he was fighting in Iraq when his first song became a hit and no one believed him when he claimed it was him.
Cyndi Lauper. Although her speaking voice is an act. Perhaps her natural voice is more like her singing voice.
Paloma Faith. Talks like a horribly twee six-year-old, belts like a diva.
Billy Idol, I was watching some show or other and the host, who was a friend of his went into where he was recording, sound checking, whatever it was. His speaking voice was…surprising and off putting. Like an adolescent frog that smoked too much. That voice haunts me to this day because he really is a good singer
I surprised when I found out Roger Whitaker has an Australian accent. Although there is one song he did where you can hear it.
Closer to south african. Born in Nairobi and lived there until he was 20 before moving to South Africa for a couple of years.
Mel Tillis!
Jason Alexander’s tenor singing voice is so good he won a Tony for Jerome Robinns’ Broadway.
And John Lithgow is a marvelous tenor.
This is an excellent example. Perry’s speaking voice is baritone. As a young man, he was able to sing powerfully into the countertenor range without resorting to falsetto.
I wonder if this also true for Geddy Lee, but I’ve never heard him talk. But I have it on good authority that it’s the case, Pavement’s Stereo:
What about the voice of Geddy Lee
How did it get so high?
I wonder if he speaks like an ordinary guy?
(I know him and he does)
And you’re my fact-checkin’ cuz
Richard O’Brien speaks in bass and sings in treble.
As a young’un in '63, I was floored when I first heard the Beatles. Then again when I heard them speak. And I felt cheated… “Hey, these guys have that cool Liverpool accent! (I didn’t know the word Liverpudlian) Why don’t we get to hear it when they sing? We just get their Americanized voices.”
I was so happy when I finally heard singers from the UK using their native accents, and I want to go back in time and give the wee Sir Paul a Proclaimers album and say “Sing like you’re proud of where you come from! We yanks can handle it!”
“Dream On” would be a good example for a spinoff thread of “What hit song sounds like it came from a different band entirely?” (See also: Foo Fighter’s “Big You”)
milli vanilli 
An example where their singing voices are nothing like their singing voices 
I remain steadfast in my belief that “Boombastic” was performed by Tom Hanks doing a Jamaican accent.
David Coverdale. After hearing Whitesnake when they were everywhere, I was really surprised when I heard his remarkably deep speaking voice in an interview.
Here’s part of an interview Dan Rather (!) recently did with him.
Well thanks, maybe Stephen Malkmus’ fact checking cuz wasn’t so great after all, cause Geddy’s speaking voice sure is quite high (though he does speak like an ordinary guy).