At the age of 21, Martin formed her own jazz quartet. In 1991, she was signed by the Scottish record label Linn Records, and her debut album, The Waiting Game, was released in 1992. The album was well reviewed and was selected by The Times as one of their 'Albums of the Year'. Martin has since released 17 CDs with the label, collaborating with musical luminaries including Martin Taylor, John Martyn, Stephane Grappelli, Kenny Barron, Richard Rodney Bennett, Jim Mullen and Nigel Hitchcock on many of these recordings. In June 2011, Martin was appointed Officer of the Order of the British Empire (OBE) in the 2011 Birthday Honours for her services to jazz.