I’ve done a bit more research on the subject and can now report my findings!
In a 2010 interview in Surviving the Golden Age, Fred says this about singing:
Your vocal style is pretty distinct. How did you develop it?
Well, I’m more of a writer. I’ve always felt my strong point is writing. That’s how it started. Keith and I used to make basement tapes; he would play guitar and I would just recite stuff off the top of my head. I can sing but I don’t know maybe I just freeze up and forget melody.
According to this short radio interview clip, posted in 2019, Fred does refer to what he does with the B-52s as “singing”, before correcting himself that he really means “talk-singing”. But then he goes on to say that now he does do some actual singing, as opposed to talk-singing:
Fred: I didn’t sing until I was in the band, really.
Interviewer: You didn’t?
Fred: No, I never sang in high school or grammar school or anything. I sang with Keith on… we had, uh, basement tapes. But, uh, was I in the chorus? No. I wasn’t going to stay after school.
Keith: I think I was the first person to hear Fred sing.
Interviewer: Really?
Keith: 'Cause I would play noise guitar and Fred would…
Fred: Recite.
Keith: Fred always had these wonderful poems, so Fred would recite—sing—his poems and I would play noise guitar.
Interviewer: And that just continued. You…
Fred: I mean talk-sing.
Interviewer: Talk-sing?
Fred: But now I sing more.
Interviewer: Yeah. Yeah, I love to hear you sing.
I was curious to hear some of this supposed actual singing, so I checked out his Wikipedia page and found that his most recent album is The Vertical Mind from 2017, issued by his side project Fred Schneider & the Superions. The album is available on YouTube. Most of the tracks I listened to featured his usual sprechgesang, though maybe a bit more melodic than with the B-52s. But at least one track does feature bona-fide singing in the traditional style. I now proudly present to you all Fred Schneider actually singing “When the Dingoes Ate the Babies”:
You’re welcome.