I suppose characterful voices that are as much the music as the narrative spring to kind.

Vin Garbutt and Dick Gaughan use their regional voices to musical effect whilst Martin Carthy's clear crisp diction manages to find warmth where many would come over cold in that style. Martin Simpson has perfected being half a beat behind his guitar to great effect. Singers such as him, Steve Tilston etc have always been good to listen to in order to learn words to songs as they play a straight bat too but you can hear the person behind the voice.

Obviously there are many who display such traits, just thought I'd mention the more well known ones to explain what I listen for.