Are there any places on the globe where magnetometers won't work?
For example, if you took a Honeywell HMC5843 to the North or South magnetic pole, would they fail to function properly?
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You wouldn't be able to use it as a compass exactly at the magnetic pole, as the magnetic field would be vertical.
You probably already know about the need to correct for magnetic declination, which is the angle between magnetic north and true north. It varies irregularly over the earth and over time. Take a look at a declination map (such as this one), and notice that there is no unique value for declination at the magnetic poles.
So the magnetometer works everywhere since it gives you the correct magnetic field X/Y/Z components; you just can't always use it as compass.
Surely they would not fail, they will show you the magnetic field just like anywhere else, it will just be quite wierd.
You still will be able to use it as a compass there if you would get approximate GPS fix to correct for magnetic pole location (but math would be way harder in comparison to usual "dumb" compass).
Justin's answer is spot on -- the magnetometer will work anywhere, but the readings may not be useful to you at the magnetic poles.
Regarding magnetic declination & inclination, there's software for a World Magnetic Model published by the US NGA & UK DGC that returns the expected magnetic fields strength & direction for a given location/date. http://www.ngdc.noaa.gov/geomag/WMM/DoDWMM.shtml
A magnetometer will 'work' anywhere there is a magnetic field that is large enough to be detected by that particular sensor. This is determined by the noise floor of the sensor, i.e. the magnitude at which the S/N ratio is large enough that the signal information can be recognized over the system noise level.
And direction matters. Even for vector sensors.
I don't think that there is anywhere on Earth, or in the Solar System inside of the orbit of Venus, where, say, a squid (or a SERF) couldn't detect a magnetic field. I think there are many fluxgates that could detect a magnetic field anywhere on Earth (which, off the top of my head, is around 0.5 Gauss - a very large value).
The important thing to recall is that there are around a dozen major classes of magnetic sensors and they all have different specs, the main one being the resolution (i.e. the smallest signal they can detect, - the noise floor).
Why do you want to find a place they don't work?