Here are all of Tom Waits' Letterman appearances.

The Bird Network Years:
1. December 21, 1983. Tom performs "Frank's Wild Years," chats with Dave, then performs another song, "The Nickel."
2. February 6, 1986. "Tango Til They're Sore," chat, and "Time."
3. October 16, 1987. Chat, then "More Than Rain."
4. October 5, 1988. "Straight to the Top" and chat.

Late Show:
5. September 27, 1999. "Chocolate Jesus" and chat.
6. May 8, 2002. "All the World Is Green" and chat.
7. September 28, 2004. Chat, then "Make It Rain."
8. November 27, 2006. Chat, then "Lie to Me."
9. July 9, 2012. Chat, then "Chicago."
10. May 14, 2015. Chat, then "Take One Last Look at the Place That You Are Leaving."
Note: The graphic card date that had Tom' first LNwDL appearance as "February 21, 1983," is incorrect; that was actually the date when Sly Stone appeared on the show. The date as I have above, December 21, 1983, was Tom's debut appearance. Notice the Christmas decorations in the studio.
What probably happened was that someone wrote down "12-21-83," but the person who typed in the date didn't see the "1" in front of the "2," thus the month February ("2") instead of December ("12").