Well in the movie Jack never sung in the No Vacancy band, he just played lead guitar for the first part of the movie before being kicked from the band. In the movie some other guy sings both songs, not sure why they'd get Jack to perform them on the OST and not mention it.
what about Elizabeth Fraser and Shara Nelson :/
She sings "Teardrop" which is probably (one of their) most famous song(s)