All credit (or blame) for this compilation goes to ‪@TweetinThies‬ and ‪@TwelveParPod‬, who both planted the seed.

1. September 17, 1985. Dave opens the show by making up new lyrics to "Memory." The clip continues for a few more minutes because.
At this stage the lyrics consist only of "Midnight, and the kitties are sleeping."

2. September 18, 1985. Dave makes a brief reference to his singing debut the night before. Later in the act, he debuts the Top Ten list, and the band closes it with a few seconds of "Memory."
Also, you can hear the band playing a few bars of "Memory" in its opening theme as Dave enters onstage.

3. September 19, 1985. The second Top Ten list, and the band again closes it with "Memory." Dave likes the concept.

For the first 20 Top Ten lists that aired from September 18 to November 4, 1985, the band played "Memory" for 14 of them. Here are three from that period:

4. September 25, 1985.
5. October 22, 1985. Preceded by TV Guide's ad for the Tri-State special. And a couple of "hypmotized" moments.
6. November 4, 1985. More "hypmotized" moments.

Note: At the end of Clip #6, the joke that Dave threatened to tell again and again until it was overheard on buses, that's here:
The Prudhomme Joke Collection, Late night, Oct.-Nov. 1985.
   • The Prudhomme Joke Collection on Lett...  

7. February 27, 1986. Five months later, Dave revises his singing career, still stuck with the opening line and nothing further.
Preceded by the Carson close voice/over.

8. November 20, 1986. A "Cats" promo, starring Chris Elliott and Gerard Mulligan.
Preceded by an actual "Cats" ad from a few years later.

9. May 21, 1987. A Tour of Hoover Dam remote aired on the last night the show was in Las Vegas. Towards the end of the piece, Dave adds more lyrics for the first time. The new version:
"Midnight, and the kitties are sleeping;
Downstairs in the basement, while birdies are cheeping."

10. June 22, 1989. Dave, Michael Keaton, and "Cats."

11. February 16, 1990. Dave, Ivana Trump, and "Cats."

12. June 18, 1991. Towards the end of a comedy bit called Guide to New York Tourists, a pre-recorded Dave sings.
Preceded by Paul's revised lyrics to Lovin' Spoonful's 1966 classic, "Summer in the City" (despite Paul sourcing it back to 1965).

Late Show:
13. August 30, 1993: Paul Newman's "Where the hell are the singing cats?"

14. September 10, 1993. Dave gives taped tour of the Sullivan Theater; a "Cats" character is outside, smoking. Also the Late Show debut of Kathleen Ankers as Helen, the Ill-Tempered Ticket Lady..

15. October 26, 1993. Roof-Cam, 1: "Cats" characters arrested on Broadway.

16. November 19, 1993. Dave thinks the show is over and taxis to "Cats."
The revised lyrics:
"Midnight, and the kitties are sleeping,
Downstairs by the fireplace, and birdies are cheep[ing]"

17. January 19, 1994. The "Cats" cast takes a curtain call.

18. August 15, 1994. Roof-Cam, 2: "Cats" characters on the garage roof. With Gerard Mulligan and Rick Scheckman.

19. September 22, 1994. Bill Murray directs the casts of "Miss Saigon," "Cats," and "Phantom of the Opera."

20. January 10, 1996. Dave invites a pedestrian onstage to be warmed up by characters from "Cats."

21. June 18, 1997. Roof-Cam, 3: Two "Cats" characters walk into Flashdancers.

22. April 23, 2008. Dave makes one more lyric revision:
"Midnight, and the kiddies are sleeping,
Downstairs by the furnace, while birdies are cheeping."

23. December 10, 2013. Brian Williams in "Cats" makeup.