Guitar
I was going to suggest Belinda Carlisle, of the Go-Go’s. She’s only credited as vocalist on all of that band’s albums, as well as on her solo albums (other than a credit for “air guitar” on her second album 
).
However, her Wikipedia entry indicates that, prior to the Go-Go’s, she was the drummer for a punk band, The Germs, in 1977.
Piano
Edit:
And Drums
Jon Anderson, formerly lead singer of Yes, played a mean tambourine, and ‘incidental’ guitar
Jon: yeah, I played incidental guitar in Yes
Toby from West Wing (channeling all other Yes members): Did you trip?
OP’s ruling: No, tambourines and other hand percussion don’t count!
Sir Mick has been playing impressive harmonica solos since the mid-sixties.
Ozzy Osborne doesn’t play instruments or write. It was actually a source of friction in Black Sabbath. He was belittled by his band mates and not considered a real musician.
I saw an interview about Black Sabbath. The friction got so bad that Ozzy was fired.
I think that is kinda funny, saying a vocalist isn’t a real musician. Last weekend I attended a fried’s operatic recital. Not at all my cup of tea, but for any less than incredible guitar picker to suggest that isn’t an amazing display of musical talent and dedication…
And, like I said, at least one school of thought has it that the lead singer/front man has plenty to do. Let them stick to their strengths, and let the musicians stick to theirs.
Hell, I see a lotta folk squawking w/ guitars around their necks who don’t impress me much w/ their skills on either voice or instrument! But many of them seem to profit nicely! 
Funny thing about Ozzie - he doesn’t sing very well either! 
Um, you mean the other musicians, right?
Ozzy played the harmonica on The Wizard.
(So is there a verdict on Brian Johnson?)
Has anyone said that? I know I haven’t.
I consider vocalists very talented musicians.
It was Ozzys band mates that ran him down. The band breakup got pretty ugly.
Oh, you were referring to the other guys in Black Sabbath, not anyone in this thread. Never mind. 
I’m not aware that Shaun Ryder plays any instruments to any decent level. I don’t recall ever seeing him do so anyway.
And as for a vocalist, well, lets just agree that his voice doesn’t count as instrument either. More like a snooker ball in a sock.
Seems like a lot of rock frontmen might be able to play the guitar, but don’t, and certainly don’t play at a professional level of skill.
As for nothing at all, I’m not so sure Henry Rollins can play any instruments. Sounds like Dee Snider might count as well.
Was she a frontman/woman?
What about Boy George (of Culture Club)?
Turntables? Or is that just a fancy tambourine?
Well, I’m pretty sure she wasn’t a frontman. I’m not sure what the exact definition is, but I’d say the Dead never really had a frontman or frontwoman.