1950s: “The Man Behind the Red Hood”
Detective Comics vol 1 #168, by Bill Finger and Lew Sayre Schwartz
Even as early as 1942, the creators of the Batman comics realized that it was easier to market their books to children if Batman's main nemesis were not a terrifying unpredictable mass murderer. As such, the Joker was gradually softened into a whimsical prankster rather than the disfigured serial killer he debuted as. By the 1950s and the advent of the Comics Code, this transformation was complete and would last for nearly thirty years.
“The Man Behind the Red Hood” is a cool little mystery story that I — or, more accurately, sixty-five years of subsequent comics continuity — have kind of spoiled the ending of by including it on this list, but c'est la vie. This story is the first to offer any kind of origin for the Joker, and later writers have come back to the well of this story again and again.
(And before any commenter offers up the comedy option and asks why it wasn't included: the infamous “boner” story is, in fact, on this list. It's “The Joker's Comedy of Errors” from Batman #66. Look, look there with your eyes: there it is.)
Many (but by no means all) of the stories on this list can be found in the Joker: A Celebration of 75 Years hardcover.
Best of the rest: “The Joker's Utility Belt” (Batman vol 1 #73), “The Joker's Comedy of Errors” (Batman vol 1 #66), “Superman and Batman's Greatest Foes!” (World's Finest Comics vol 1 #88), “The Joker's Millions” (Detective Comics vol 1 #180), “The Man Who Wrote the Joker's Jokes” (Batman vol 1 #67), “Crime-of-the-Month Club” (Batman vol 1 #110), “The Joker's Crime Costumes” (Batman vol 1 #63), “The Crazy Crime Clown!” (Batman vol 1 #74), “The Crimes of Batman” (World's Finest Comics vol 1 #61)