The singularity expansion method has been extensively applied to electromagnetic scattering problems for target identification based on the aspect-independent natural frequencies. This chapter aims to extend the residue decomposition as done in electromagnetics to acoustic and elastodynamic pole residues. The resulting factored form of the residues shows that there are two angular functions, one scalar and one vector, that characterize the elastodynamic residues. In the electromagnetic context, the properties of the residue dyadic have been determined from an integral equation for the currents on the scatterer. The electromagnetic singularity identification and magnetic singularity identification have their various advantages and limitations, depending on the characteristics of the target and the surrounding medium. Another technique with yet different advantages and limitations involves the scattering of sound waves from the target.