The possible presence of a real discrete spectrum of eigenfrequencies is considered for a vibrating inhomogeneous membrane of infinite extent. The membrane contains a distributed mass in the form of a layer of finite length oriented along the infinite membrane dimension with no stiffness at the boundary between the membrane and the layer. In contrast to the continuous spectrum describing the waves propagating in the membrane, the discrete spectrum determines the nonpropagating modes of vibration (the trapping modes) localized near the inhomogeneity and characterized by an amplitude exponentially decreasing at infinity. The discrete spectrum is finite and occurs below the first cut-off frequency. The possibility of the extension of the discrete spectrum to the region of the continuous spectrum is analyzed: the spectral problem is considered with the added condition that determines the occurrence of a discrete spectrum above the first cut-off frequency. The approximate solution to this problem points to the absence of a discrete spectrum in the aforementioned region.