The problem for me is that vocals are a big part of the music, and even if the singer is a "weak singer", if the vocals fit, it's cool. Jane's Addiction, Motorhead.
So I've answered this about bands I SHOULD like, but the singer is the last hurdle and I haven't cleared it yet; I would add Red Hot Chili Peppers to the list, and Motley Crue (especially live). I saw the Dr. Feelgood tour, and musically it was solid as a fucking rock (Tommy is a machine) but I just got tired of every line being "...........................................second class hood!..................................Hollywood!" in that pitchy, whiny voice of Neal's. Man, does he blow.
I like it, because she doesn't sing much, but Donna Godchaux belongs here; I have a ton of Dead from '74 to '78, and it's almost all excellent, unless and until she comes in wailing like someone stuck a wah wah pedal in her ass.