The problem isn't that Cohen shouldn't be allowed to *sing* his own songs. He has a great baritone croon. The problem is that he shouldn't be allowed any input into how they are orchestrated. The version I linked to does feature addition verses, which hit harder on the sacred/profane/prayer/sex theme than the verses in the popular covers.
The Jeff Buckley version is one of the most popular. It has some fucking incredible guitar work, which makes for a better arrangement than the piano arpeggios that seem to be standards for covers Hallelujah. This is actually my first experience listening to Buckley, despite reading about him constantly.
Buckley has a verse not in Shrek: "Remember when I moved in you/and the holy ghost was moving too/ and with every breath we drew was hallelujah" I swear Cohen has a way of making his own horndogginess into something transcendent. I wish my horndogginess were so transcendent.
The standard arpeggio covers are here (Rufus Wainwright) and here (Alison Crowe, which I only clicked on because I thought it said Alison Krause)
Its a close contest between Buckley and Cale.