The myth goes that if you flush a toilet in Australia the water swirls down the drain the opposite way than in the northern hemisphere, due the Coriolis effect (an apparent force which describes how objects veer to the left or right when traveling on something that’s rotating — see the link above for a good visualization of this).
If there were no other forces on that water in the sink or toilet, that would be true. The Coriolis effect does actually make hurricanes rotate the opposite direction in the two hemispheres. But for toilets and sinks it’s another story. The toilet myth is easy to dispell — just peek around the rim of the toilet and you’ll see that the water is jetted into the bowl at an angle, which determines the direction the water swirls. Sinks, however, are a little more tricky.
I’ve heard of charlatans who hang around the equator in Kenya, carrying basins of water. They’ll stand on the southern side of the equator with the basin, pull a plug at the bottom, and show that it swirls out counter-clockwise. Then they’ll walk to the northern side of the equator, fill the basin and pull the plug, and it swirls out clockwise. Irrefutable proof? Be careful! You have to know all the initial conditions in any experiment, and in this one, there is one that is hidden from you. The huckster just has to add a slight rotation to the water before letting it out (for example, pour the water in at a very slight angle to give it an initial rotation, and it will “remember” that rotation as it swirls out of the basin. In fact, you can swirl the water in the basin, then walk away from it for several hours, and it will still “remember” that rotation when you pull the plug! Plus, the charlatans got it backwards — water should actually swirl counter-clockwise in the northern hemisphere if the Coriolis effect were at play! (See Alistair Fraser’s website for a great explanation of how you can re-create this fakery for a fun party trick!)
Even if you don’t give the water an initial swirl, tiny pits and imperfections in the basin can give the water a rotation — which may be clockwise or counterclockwise, but doesn’t depend on which hemisphere you’re in!
Now, all that said, the Coriolis effect does play a role. It’s just really tiny relative to all these other effects. It’s about 10 millions the size of the effect of gravity. So, if you have a perfect basin, with completely still water, then the Coriolis force will make the water swirl opposite directions in the two hemispheres. This was demonstrated by Ascher Shapiro, a researcher at MIT in 1962. You can see the Straight Dope talk about this topic, or detailed information from Alistair Fraser.