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The separation-of-variable (SOV) methods, such as the improved SOV method, the variational SOV method, and the extended SOV method, have been proposed by the present authors and coworkers to obtain the closed-form analytical solutions for free vibration and eigenbuckling of rectangular plates and circular cylindrical shells. By taking the free vibration of rectangular thin plates as an example, this work presents the theoretical framework of the SOV methods in an instructive way, and the bisection–based solution procedures for a group of nonlinear eigenvalue equations. Besides, the explicit equations of nodal lines of the SOV methods are presented, and the relations of nodal line patterns and frequency orders are investigated. It is concluded that the highly accurate SOV methods have the same accuracy for all frequencies, the mode shapes about repeated frequencies can also be precisely captured, and the SOV methods do not have the problem of missing roots as well.