Newbie here! (Although I was a fan back at the beginning.) I've been catching on on All Things Cohen in the last few weeks but my current favorite is Everybody Knows. I feel in love with the video even though it is of poor visual quality. I guess it corrupted over time.
http://www.youtube.com/watch?v=wh9AC0jCGjY It still sounds great, though!
Other songs that I can't currently get out of my head are Dance Me to the End of Love.
[small voice]His shared glances with the back-up singers in the video are enough to...um, melt me. Hey, he's a handsome man and he brings out my inner fangirl![/small voice] 
Is Jennifer Warnes (sp) one of the singers? Also Closing Time.
His early songs with always have a warm place in my heart. Suzanne, That's no way to Say Goodbye, Sisters of Mercy, Bird on a Wire. Joan of Arc.