A cat riddle is:
 
Q: What do cat actors say on stage
A: “Tabby or not tabby!”
 
This joke—a parody of the “To be, or not to be” line from William Shakespeare’s Hamlet—has been cited in print since at least 1999. “Tabby, or not tabby—that is the question!” was in the play Whittington, Junior, And His Sensation Cat: An Original Civic Burlesque (1870) by R. Reece.
 
     
Wikipedia: To be, or not to be
“To be, or not to be” is the opening phrase of a soliloquy spoken by Prince Hamlet in the so-called “nunnery scene” of William Shakespeare’s play Hamlet. Act III, Scene I.
 
Though it is called a soliloquy Hamlet is not alone when he makes this speech because Ophelia is on stage pretending to read while waiting for Hamlet to notice her, and Claudius and Polonius, who have placed Ophelia in Hamlet’s way in order to overhear their conversation and find out if Hamlet is really mad or only pretending, have concealed themselves.
         
Google Books
Lacy’s Acting Edition of Plays, Dramas, Farces and Extravagances, Etc., Etc:
As Performed at the Various Theatres ...
By Thomas Hailes Lacy
London, UK: T. H. Lacy
1849 (This date is possibly incorrect. This is probably from 1870. The play appears to be Whittington, Junior, And His Sensation Cat: An Original Civic Burlesque by R. Reece.—ed.)
Pg. 27:
MARINER. Tabby, or not tabby—that is the question!
 
Chronicling America
18 December 1880, Public Ledger (Memphis, TN), “Stray Shots,” pg. 1, col. 5:
Tabby, or not Tabby, that is the question of the cat crusade.
       
Google Groups: alt.games.tiberian-sun
non-violent cat jokes
Walrus Boy Bigger and better
10/23/99
(...)
What do cat actors say on stage ?
Tabby or not tabby ! 
 
Google Groups: alt.humor.puns
Puns of the Weak 02/15/02
Stan Kegel
2/15/02
(...)
What do cat actors say on stage?
Tabby or not tabby!  (Douglas Helsel)
   
Google Books
Laughs
By Matt Rissinger and Philip Yates
New York, NY: Sterling Publishing Company, Inc.
2004
Pg. 237:
Teacher: What’s a cat’s favorite quote from Hamlet?
Janet: “Tabby or not tabby!”
 
19 February 2006, Sunday Telegraph (Surry Hills, New South Wales), “The Funday Telegraph” by Richard Clune, pg. 17:
What do cat actors say on stage?
Tabby or not tabby.
   
Twitter
❤️ FizzyDuck ❤️
@FizzyDuck
What do cat actors say on stage?Tabby or not tabby.
1:43 PM - 25 May 2009
 
Twitter
Lucky Score
@LuckyScoreNet
What do cat actors say on stage ?
Tabby or not
tabby !
#Joke from #CatJokes in #Jokes, #Humor and #Satire
12:22 AM - 9 May 2018