The main aim of this paper is to examine the efficiency and the effect of various single-term weighted residual methods, including Bubnov–Galerkin, least squares, and point collocation methods on the free vibration behavior of viscoelastic plates. The refined classical plate theory and Kelvin–Voigt/standard solid viscoelastic models are adopted for kinematic and constitutive relations of foam plates, respectively. The spatial domain is discretized using weighted residual methods with shape functions of bending component of transverse deflections. The resulted algebraic eigenvalue problems with frequency-dependent coefficients are solved via an iterative numerical algorithm. For elastic plates, the present results based on four different methods are compared with their counterparts which are obtained based on 3-dimensional, classical, first- and higher-order shear deformation theories under Navier, Levy, and fully clamped boundary conditions. For simply supported viscoelastic plates, frequencies are compared with exact solutions, and acceptable accuracy is observed. Then, parametric studies are undertaken to assess the vibration mode, bulk/shear ratio, thickness ratio, material model, and boundary condition. From these results, it is revealed that bulk–shear ratio is a key factor while material model makes no significant difference on the real part of the vibration frequencies of fully free plates.