Why GPS is at LEO? Why GPS/GLONASS/Galileo satellites are on low earth orbit?
Why geostationary orbit is so bad? Sattelites might be placed there 'statically' and more precise...
The only problem I can see is navigation close to poles, but they have this problem anyway.
 A: This question always interested me.
I found this recently -
"Another issue we wrestled with is which satellite orbits to use. We did not want to be in geostationary or geosynchronous orbits. The reason was these alternatives would force us to deploy ground stations on the other side of the globe, whereas, by putting them in some orbit that periodically passed across the United States, you could update the knowledge of where they were and what time it was on the satellite, then store that information in the satellite and continue to broadcast as it went around the Earth. That is the fundamental way we ended up with twelve-hour orbits. We also wanted to be reasonably high because we didn’t want the orbits significantly disturbed by the atmospheric drag. At the same time, by going high you had more visibility, more coverage on the Earth. So with an Earth coverage antenna and suitable power densities on the Earth, you ended up with the ability of twenty-four satellites to provide very solid, total Earth coverage."
Brad Parkinson
See:
http://www.ieeeghn.org/wiki/index.php/Oral-History:Brad_Parkinson#Parkinson.27s_educational_background_and_the_origins_of_GPS.5D
A: This question is slightly faulty...  Part of the US GPS system is geostationary (the WAAS component).  It's used in conjunction with the non-geostationary birds for higher precision fixes.  While primarily used for aircraft instrument approaches, there are off the shelf USB GPS computer peripherals that use WAAS.
A: The Chinese satellite navigation system is in geostationary orbit.  From the Wikipedia article:

Unlike the American GPS, Russian GLONASS, and European Galileo systems, which use medium Earth orbit(MEO) satellites, BeiDou-1 uses satellites in geostationary orbit(GEO). This means that the system does not require a large constellation of satellites, but it also limits the coverage to areas on Earth where the satellites are visible. The area that can be serviced is from Logitude 70°E to 140°E, and from Latitude 5°N to 55°N.

A: Why they're not on the geostationary orbit?
It's because the geostationary orbit - and indeed, there's just one such orbit, a one-dimensional curve in space - only exists above the equator while the GPS has to cover the whole planetary surface, including the points closer to the poles.
However, it's untrue that the GPS satellites are located at the LEO, either. Low Earth Orbit is defined as 160-2000 kilometers of altitude. However, the GPS satellite constellation is located roughly 20,200 km above the surface - over one half of the geostationary radius - in such a way that the position of each satellite returns to the same place twice per 24 hours. This is very convenient for synchronization and planning.
The Galileo satellites will be at altitude 23,222 km. It's also an intermediate circular orbit, much like for the 19,100 altitude of the GLONASS whose orbital period is about 11 hours.
