I decided to put a list of released songs in which Stewart sings more than background (or at least where he chants in the case of Wield the Spade.)
Did I miss any songs?
Police:
On Any Other Day
Does Everyone Stare (first verse)
Miss Gradenko
Klark Kent (some of which were released as SC rather than KK songs):
Too Kool To Kalypso
Strange Things Happen
Thrills
Excesses
Love Lessons
Office Girls
Away From Home
Don't Care
My Old School
Ritch In A Ditch
Stay Ready
Guerilla
Yo Ho Ho
From the Rhythmatist:
Serengeti Long Walk
African Dream
Gong Rock Remix
Oysterhead:
Wield the Spade
[ETA: added Does Everyone Stare]
[ETA2: added Miss Gradenko after the discussion here]