Waterfall Into the Abyss

An almost obligatory feature of Floating Continents, Floating Castles, and Flat Worlds, (especially if the setting is Crystal Spires and Togas) is waterfalls off the side of the structure. Awesome but Impractical in the case of smaller floating islands since you'd expect the water to run out fairly quickly since they don't have a very large area of land from which to collect rainfall. It's especially strange in the case of Flat Worlds since it doesn't seem that there would be anything to bring the water back.

Examples of Waterfall Into the Abyss include:

Anime and Manga

In One Piece, during the CP9 arc, the main judicial building Robin is being held in is on a island hanging in the middle of an abyss by a narrow section of land far too frail to actually support it. The abyss is in the middle of the ocean, which drains eternally into this bottomless pit.

Film

The Hallelujah Mountains in Avatar.
The titular location in Pirates of the Caribbean: At World's End.
The edge of Asgard in Thor.

Literature

The world of The Chronicles of Narnia has this feature described in some detail in Voyage of the Dawn Treader. Continually sailing to the east doesn't bring you around to the west again, but to The End of the World. If you go over the edge, you end up in Aslan's Country—a kind of Fluffy Cloud Heaven, since most people who arrive there die first.
Terry Pratchett's Discworld has the ocean falling off all sides of the Disc, but "arrangements are made" (it's probably quantum). There are even people who take advantage of this and have put a net around the edge (the "circumfence") to catch floating items for salvage.
- His earlier work Strata had a different disc-world made by Ancient Astronauts terraformers, which (IIRC) had a hidden system to pull the water back up.

MMORPGs

Aion has some of these in the aptly named Abyss.
Outland in World of Warcraft has both small floating islands with waterfalls on them, as well as a few places where water falls off the edge of the world.
- Also Necropoli and their derivatives in World of Warcraft and Warcraft III has slime waterfalls flowing out of them.
Wizard 101 has a lot of these because of the Shattered World setting.

Video Games

Chrono Trigger's Kingdom of Zeal, of course. Oddly enough, walking around on the surface of 12,000 B.C. you can't find any torrential downpours falling out of nowhere... so probably the wizards living on the floating continent have something worked out.
This appears in Golden Sun, which has a Flat World.
The Myst novel The Book of Atrus featured something like this in a small, gravity-defying Age that Catherine made. Supposedly the water falling off into the abyss turned to vapor almost instantly and rose to the top of the world, where it condensed and rained back down again, perpetuating the cycle. Atrus was having fun working out the physics behind it, Catherine just thought it was cool.
The Beach Bowl Galaxy in Super Mario Galaxy has this. On multiple levels.
One puzzle in Banjo-Tooie involves removing the plug from a pool of water in Cloudcuckooland.
These are all over the place in Skies of Arcadia, where The Sky Is an Ocean but most every Floating Continent has a river or two.
The vaguely Sonic-like and so-cute-you'll-vomit SNES platformer Rainbow Bell Adventure had plenty of floating islands with waterfalls in the background.
May appear on floating islands in Minecraft and can be intentionally created by the player using a water or lava "source block".
Subnautica's "Lost River" biome -- an undersea cave system -- features an unusual variant, in which currents of poisonous green and blue liquids flow as rivers and waterfalls separate from the water around them, with some falls that plummet through the sea floor to unknown depths below,
Ouroboros in City of Heroes was situated on a flying/hovering base; surrounding it were six islands/mountains also hovering in midair, from which waterfalls fell into the ocean below.

Western Animation

Appears in the trailer for Disney's version of Sinbad.