"Goodbye Cruel World" by Shakespear's Sister

Shakespear's sister was a duo that formed in the late 80s and released one of the most critically acclaimed records of the early 90s. Their sophomore effort, "Hormonally Yours," yielded their biggest single in the US, 1992's "Stay," which soared to #4 (if you've never played the acoustic version, go do it now). 

Their excellent follow-up single, "I Don't Care," stalled at #55 for some reason - it deserved much better, as did many other tracks from this record, including "Goodbye Cruel World."

The group was comprised of Siobhan Fahey and Marcella Detroit, who was known earlier as Marcy Levy. Both had strong pedigrees: Fahey was in Bananarama and was married to Dave Stewart of the Eurythmics, who co-wrote "Stay." Detroit co-wrote and sang backup on Eric Clapton's hit "Lay Down Sally," and released a solo debut record in 1982.

Detroit was fired in 1993 and Siobhan cancelled the project in 1996. They revived things in 2009, but have yet to get back on the charts.