The paper is concerned with the localization phenomenon in continuous structures of finite and infinite length. An example of a compressed infinite beam is used to investigate the localization of oscillations in the area of a defect in the foundation and to examine the features of buckling of the structure for this case. It is shown that, in addition to the continuous spectrum, the existence of trapped modes is related to the appearance of a point (discrete) spectrum below the cut-off frequency of the structure. The dependence of the localized point frequencies on the compressive force is obtained. The convergence of the fundamental frequency to zero with increasing force specifies the localized buckling mode and the critical force, which agree with the solution of the corresponding static problem. The obtained results are compared with the results available for finite systems.