Spectral representations for two-point functions of point quantum-mechanical systems are derived, under very general assumptions. Sum rules are then obtained, both for systems with velocity independent and dependent interactions. The implications of these sum rules are illuminated on particularized simple systems. Special attention is paid to systems where a zero-frequency (translational) mode exists. The difference between velocity-dependent and -independent interactions is shown to be very crucial in this respect. The same analysis provides an instructive example of inadvertent misuse of the spectral representations. Finally, the possibility of Goldstone behavior, with all its consequences and implications, is demonstrated for systems with finite many degrees of freedom.