Some of these, it's hard to say, because it can be chalked up to "that's exactly what they're going for." Someone mentioned Marilyn Manson, who... I don't know if anyone would call him a "great" singer, but his vocals are absolutely part of his draw. He basically has a selection of voices (abrasive singing, screaming, deep croak, falsetto, hissing whisper) and he changes between them like it's a separate instrument.

I'd put Bob Dylan in that same boat. There's a folksy charm to his voice that's absolutely part of his schtick, but I'm in the camp that says the best version of a Bob Dylan song is when it's by someone else.

My submission would be Cake (dunno if they're still famous. Remember Going the Distance?). There were a number of bands in the 90s who didn't really sing, they kinda talked their way through their songs. Soul Coughing was another one. I liked them well enough, probably because I could actually understand what they were saying. I just hope the singers were also playing an instrument, because everyone else was working real hard.