An accurate method is presented for flexural vibrations of rhombic plates having all combinations of clamped and simply supported edge conditions. A specific feature here is that the analysis explicitly considers the bending stress singularities that occur in the two opposite, clamped-hinged and/or hinged-hinged, corners having obtuse angles in rhombic plates. The strength of these singularities increases significantly as the obtuse angles at the clamped-hinged and/or hinged-hinged corners increase. Stationary conditions of a single-field, classical thin-plate Lagrangian functional are derived using the Ritz method. Assumed transverse displacements are constructed from a hybrid set of (i) admissible and mathematically complete algebraic polynomials, and (ii) comparison functions (termed here "corner functions") which account for both the kinematic boundary conditions and the bending stress singularities at the obtuse clamped-hinged and/or hinged-hinged corners. Extensive convergence studies demonstrate that the corner functions accelerate the convergence of solutions, and that these functions are required if accurate solutions are to be obtained for highly skewed plates. Accurate non-dimensional frequencies and normalized contours of the vibratory transverse displacement are presented for rhombic plates having a large enough skew angle of 75° (i.e., obtuse corner angles of 165°), so that the significant influence of the corner stress singularities may be clearly understood. Accurate solutions for isosceles triangular plates with various combinations of clamped-hinged edges are also available from the frequency and mode shape data presented. The upper bound frequency solutions derived from the present analysis are shown to improve upon the existing upper bound results in the published literature.