This article is the first to classify the phenomenon of trapped modes initiation in elastic systems with mass--spring inclusions. Necessary and sufficient conditions for low- and high-frequency trapped modes were found. It is shown then that in elastic systems (beams, plates, shells) which are bounded by a continuous spectrum displaced from 0 and have mass inclusions, a discrete spectrum of natural frequencies occurs before the first boundary frequency. Trapping modes of oscillation corresponding to those frequencies possess a particularly pronounced local character in the inclusions region. The discrete spectrum displacement to the continuous spectrum axis and initiation of high-frequency trapped modes are accompanied by a phenomenon of traveling waves damping (antiresonance).