We study the general scattering interaction of electromagnetic (EM) pulses of arbitrary shape and duration with a spherical target. The target is assumed penetrable and we model it either as totally dielectric, or as perfectly conducting but covered with a thin outer masking coating of a dielectric material. We obtain the radar cross-sections (RCS) of such targets and analyze the many resonance features that are present within their resonance region. The dielectric composition makes the resonance features become very prominent and it relates them to the eigenfrequencies in ways analogous to those of the Singularity Expansion Method (SEM), originally developed for perfectly conducting scatterers. Transient echoes from these targets are linked to poles and residues in the complex-frequency plane. The individual resonances associated with each pole (i.e., eigenfrequencies) can be studied one at a time, provided we use long illuminating pulses since these excite transients at their carrier frequencies that ring and decay. Of greater importance is the use of short pulses, since these are shown to replicate the entire RCS of the target in bands that have widths directly proportional to their energy and carrier frequency. We develop a methodology that can handle pulses of any conceivable spectra, interacting with (lossy or lossless) dielectric scatterers, and predict the backscattered returns. We illustrate it for various types of scatterers' compositions and/or pulses, with displays in the frequency and time domains, and give physical interpretations of the results.