As we now know from the season 2 premiere of “True Detective,” every character is carrying some bleak, heavy baggage—including that hollow-eyed woman singing to a near-empty dive bar about broken wings and a life up in smoke.

“She has clearly not made the best choices in life, based on the way she looks and where she is playing her music,” says Lera Lynn, who plays the nameless chanteuse. Slouched on a stool, she performed solo with an electric guitar, as self-destructive detective Ray Velcoro (Colin Farrell) and self-doubting...