On her first album, released on the influential él label, Griffin had no band of her own and was backed by members of The Monochrome Set. "The Camera Loves Me" was critically acclaimed in the UK and Japan. In 1993 Griffin worked with the Monochrome Set on a second album, 'Mondo', produced by Monochrome Set singer Bid and released on the Japanese label Polystar. This album was later released in the UK on the Cherry Red label.
Peter Momtchiloff, formerly of Talulah Gosh, Heavenly, and Marine Research, joined the band as lead guitarist and bassist in 1999. Two EPs, "Emmanuelle Béart" and "Sugar Mummy", were released in 2001, followed in 2002 by a new album, "Brief Lives", a joint release by US label Matinée Recordings and UK-based Fortuna Pop!. By this time two new members had joined the band: Deborah Green (former member of Thee Headcoatees) on drums and Lupe Nuñez-Fernandez on bass. After the band recorded a fourth album, "The Morning After", in 2004, Nuñez-Fernandez left to concentrate on touring and recording with her own band Pipas and was replaced by Andy Warren, former member of Adam and the Ants and the Monochrome Set. Since then, the Would-be-goods have recorded a fifth album which is due to be launched in October/November 2008.
The Would-be-goods have performed live in the United States, Spain, Sweden, Finland and Denmark as well as the UK.