Asymptotic expansions are constructed for the eigenvalues and eigenfunctions of the Dirichlet problem for the biharmonic operator in thin domains (Kirchhoff plates with clamped edges). For a rectangular plate the leading terms are asymptotically determined from the Dirichlet problem for a second-order ordinary differential equation, while for a T-junction of plates they are determined from another limiting problem in an infinite waveguide formed by three half-strips in the shape of a letter T and describing a boundary-layer phenomenon. Open questions are stated for which the method developed gives no answer.