QUESTION:
How deep do the ice caps go into the ground, and how thick are they?
 
ANSWER from Ken E Herkenhoff on December 11, 1996:
There are two types of ice caps on Mars: the seasonal carbon dioxide
("dry ice") frost caps and the permanent ice caps.  

The seasonal caps, which grow in the winter to an average depth of 
about a meter, completely evaporate in the northern summer to expose 
the permanent water ice cap. 

The water ice cap overlies the polar layered deposits, which are thought
to be composed of a mixture of dust and water ice and may be several
kilometers thick.  The thickness of the permanent ice cap is not known,
but it must be at least a few centimeters thick to be as bright as it is. 

The south polar cap is apparently always covered by carbon dioxide frost,
but also overlies polar layered deposits of dust and water ice.  The south
polar layered deposits do not appear to be as thick as the north polar
deposits, but the topography (relative elevations) of the region is not
well known.  The laser altimeter on the Mars Global Surveyor mission will
hopefully return accurate topographic data that will be useful in
constraining the thickness of the layered deposits at both poles.