What could cause taps for both hot and cold water to suddenly stop working?
The pipes are not likely to be blocked. The problem will most probalby be in the tap. Most taps usually have a aerator before the exit and that could be blocked up from dirt.

To clean the tap there is a tool to remove the aerator but it is not necessary. You can use pliers or a the correct size key.


There is also a helpful video here

And this is what the part looks like when its a bit clogged.



With the clog that you mentioned it must be pretty bad- So when you remove the aerator you will most likely see allot of dirt follow it. Open the taps for a minute to get it out of the system.

This dirt usaully comes from the main water supply so installing a filter for your main supply for the house if a good option to stop this nuisance. There also options to improve the drinking qualities. Ask you local plumber depot for advice.

DIY TIPS


To avoid damaging the aerator body it is a good idea to wrap it with some cloth and gently try to unscrew it. It could be a bit stuck but wiggle it back and forth and should let go easily. 
Many aerators  go into several pieces. The can help in cleaning them. Try to remember the order.
The aerator is not mandatory. It helps make the water come out silky and uniform which helps to prevent splashing. You can use the tap without it if it clogs up frequently.