I am keeping a list of songs that I hear and like and would like to hear Morrissey cover.
The list will be ever expanding, but ever so slowly.
This list so far (in no particular order):

The Sundays "You're Not The Only One"
Depeche Mode "Everything Counts"
Depeche Mode "Deaths Door"
Mazzy Star "Give You My Lovin"
Mazzy Star "Blue Flower"
10,000 Maniacs "Verdi Cries"
Dead Can Dance "Fortune Presents Gifts Not According To The Book"
The Innocence Mission "Notebook"
This Mortal Coil "Til I Gain Control Again"
Nick Cave "The Weeping Song"
Nick Cave "Into My Arms"
The Sisters of Mercy "More"
The Cranberries "Linger"
Leonard Cohen "Heart With No Companion"
Leonard Cohen "Famous Blue Raincoat"
Leonard Cohen "The Window"
Heather Nova "Maybe An Angel"
Love & Rockets "Haunted When the Minutes Drag"
Pet Shop Boys "It's A Sin"
New Order "Regret"
REM "Everybody Hurts"