There is more than one artist with this name:

1) Blue Murder was the hard rock brainchild of ex-Whitesnake and Thin Lizzy guitarist John Sykes. The original lineup of the band, founded in the late-80's, also included ex-The Firm bass player Tony Franklin and drummer Carmine Appice.

Hoping to capitalize upon the success of Whitesnake's multi-platinum 1987, Geffen Records signed Sykes to form a new band. After auditioning several vocalists (including Tony Martin and Ray Gillen), Sykes was encouraged to handle vocal duties himself.

Their self titled debut did moderately well, peaking at #69 on the Billboard Top 200, and spawned the singles "Jelly Roll" (#15, Billboard Mainstream Rock Tracks) and "Valley of the Kings", which received substantial airplay on MTV's Headbanger's Ball.

In 1993, the follow-up Nothing But Trouble was released. While the original lineup recorded much of the album, Sykes' hesitancy at handing the vocal chores resurfaced; he recruited Baton Rouge singer Kelly Keeling. After spending more than a year laying down the vocal tracks for the album, John was not convinced that the vocals were to his liking. He proceeded to go over the lead vocal tracks with his own until Kelly had only one lead vocal track left, "I'm On Fire".

Perhaps stemming from Keeling's ouster, Franklin and Appice left midway through the recording sessions, to be replaced by bassist Marcos Mendoza and drummer Tommy O'Steen.

As Nothing But Trouble was released well after the grunge explosion of 1991, it received virtually no promotion from Geffen. The album's lead-off track "We All Fall Down" reached #35 on the Billboard Mainstream Rock Tracks chart, but the album was a commercial failure.

A live recording, Screaming Blue Murder, was to be the band's final release.

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2) Blue Murder was a Dutch band formed after the split up of Soviet Sex back in 1982. The members on the first albums are:
MAARTEN VAN DER PLOEG: voc, guitar
PHIL VAN TONGEREN: voc, synthesizer
RYU TAJIRI: voc, synthesizer
ROGIER VAN DER PLOEG: guitar
ANNEMIEK LELYVELD: bass
MONIEK VOULON: drums

Also on drums on the latter albums: Jos den Brok, and in 1986 Kors Eijkelboom. In 1987 the group falls apart. In the Amsterdam Melkweg arena was their final concert.
Blue Murder plays guitarpop, like Talking Heads.

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3) Blue Murder is an occassional folk supergroup, containing various members of Swan Arcade, Coope Boyes and Simpson, Waterson:Carthy and The Watersons.