I would like to know the name of the singer and the title of the song.
The only clue that I have is lyrics "that wild, destructive thing
called love." It is from a play "A Taste of Honey" by Shelagh Delaney.

I found that you can buy the book of the same name and by the same
author. The link is http://www.amazon.com/exec/obidos/tg/detail/-/0802131859/ref=pd_sim_books_1/104-0686508-4542338?v=glance&s=books

Hm, I don't have my copy of A Taste of Honey handy at the moment so I
can't check that line to see if it's worded slightly differently.   
The Smiths/Morrissey have taken lots of lines from that play (both the
play and film versions), but I haven't heard that specific line in a
Smiths song before.

If you'd like to see all the cases in which Morrissey took lines from
the Taste of Honey, check out the following page.  The link is the
cached version so that you can quickly spot the highlighted Taste Of
Honey/Shelagh instances:
http://216.239.57.100/search?q=cache:rwWb7oa00moJ:www.compsoc.man.ac.uk/~moz/nicked.htm++Shelagh&hl=en&ie=UTF-8

And slightly related, Shelagh is also the person on the cover of the
Smiths' Louder Than Bombs album.