Having done some work at a major research institute I've had the opportunity to check out their mobile field lab department, which is equipped to deal with that sort of situation.
Their mobile unit looked quite impressive, a modified army truck transformed into a small lab on wheels. I remember they had a number of suitcase-sized kits containing various reagents for detecting specific kinds of chemical contamination. IIRC, several such kits would be needed in an event of heavy contamination from unknown sources. There was also a portable XRF device used as a quick but not very precise means of identifying some types of contaminants (heavy metals probably).
It went well beyond my knowledge of chemistry but I was quite impressed by the amount of gear they had available. Even more so when I learned that it was only good enough to carry out very basic field analyses. For really accurate results samples would need to be examined in a proper lab back at the institute.
So I wonder what an average person could do when in doubt about contaminated water. Probably very little. It seems that even distillation cannot remove all the typical contaminants:
"Removal of organic compounds by distillation can vary depending on chemical properties of the contaminant. Certain pesticides, volatile solvents, and volatile organic compounds (VOCs), such as benzene and toluene, with boiling points close to or below that of water will vaporize along with the water as it is boiled in the distiller. Such compounds will not be completely removed unless another process is used prior to condensation."
http://www.ianrpubs.unl.edu/epublic/live/g1493/build/#target2