AFM based single molecule force spectroscopy was used for the investigation of single polyelectrolyte chains. Namely, the elasticity of polyvinylamine chains and their desorption from solid surfaces was studied as a function of the polymer's charge density and electrolyte concentration. Experimental force-distance profiles were fitted by the wormlike chain model, including elastic contributions arising from the stretching of bond angles and covalent bonds. It was found that, under the high stretching forces which can be applied in the AFM experiments, the bending rigidity of polyelectrolyte chains (as described by the persistence length) is significantly lower than predicted by Odijk-Skolnick-Fixman (OSF) theory. Furthermore, the desorption force of single physisorbed polymer chains from negatively charged silica surfaces was determined. In addition to the electrostatic interaction between polymer and substrate, which depends linearly on the Debye screening length and the polymer's line charge density, a constant nonelectrostatic contribution to the desorption force was observed.