Actually, moving water (like the waterfall in this photo) can freeze. But it has to be a lot colder to do it.
To understand this, you can think about the way water freezes. When the water changes from a liquid to a solid, it starts in just one place, with just a couple of water molecules getting stuck together (because they're cold and not moving around very quickly). Then another molecule sticks to the first two. And another one sticks to those three, and soon you end up with a huge ice crystal that just keeps getting bigger and bigger as more water freezes onto it.
In order for the first two water molecules to stick together, they have to be pretty close together and not move away from each other. If the water is standing still, this is pretty easy - as the temperature drops, the molecules stop moving around as much, and they start sticking together. But if the water is moving, like in a waterfall, the molecules are more likely to move away from each other and less likely to get stuck together. So the temperature has to get a lot lower in order for the first ice crystals to form. But once you've got the first few, the rest will freeze more easily.
This is why they say that you should run the water in your sink (just a little) if it gets really cold outside. If the water is running, then it's moving through the pipes instead of sitting still. (Also, the moving water will provide some heat that will also help prevent freezing).
(published on 10/22/2007)