One thing that pops up regularly is "the 1920-21 depression was very deep but we got out of it quickly!"
But if you compare that to the duration of recessions in the period after the Federal Reserve began operations (in 1914), it is tied as the second longest recession on record. And which episode is it tied with? You guessed it! The current "Great Recession" - which has also been dated as lasting 18 months.*
Now, how much do these durations really mean? That, of course, is up for debate. One of the biggest reasons why we even care about economic fluctuations is the impact they have on unemployment. Unemployment can persist long after the decline in economic activity has ended, and we've certainly had periods where growth has picked up but employment hasn't. It's legitimate that people aren't so sanguine about such occurrences. How you rank recessions based on what matters to you is your business. But the point is, under no reasonable definition was 1920-21 a quickie. Even if you redefine what matters to you so that 1920-21 is no longer tied in second place, it's still among the longer downturns in recent American economic history.
UPDATE: That's a lie... I'm not sure why I said that in the second paragraph. I know exactly where this meme got started - the same place that most of the memes about the 1920-21 depression that are floating around there got started.
*The 1913-1914 recession was longer than 1920-21, but it was well under way by the time the Federal Reserve started operations, which is why I left it out of my "post-Fed" count. This recession endured 12 months after the passage of the Federal Reserve Act in December of 1913. I'm not sure when Fed operations started in 1914, so I'm not sure how long it lasted after the Reserve System itself got off the ground.