This analysis deals with free vibrations of a rectangular plate with a side crack by using the famous Ritz method with special displacement functions. To appropriately describe the stress singularities at the crack tip and show the discontinuities of displacement and slope crossing the crack, previously used corner functions, as well as a new set of functions, are added to the well known admissible functions consisting of regular polynomials. Comprehensive convergence studies on the vibrations of simply supported rectangular plates with horizontal cracks at the symmetry axis are carried out and show that both the corner functions and the proposed new set of functions indeed accelerate the convergence of numerical solutions. Furthermore, the new set of functions is found to be particularly capable in improving convergence of solutions, especially when there is a large crack. Convergence studies also demonstrate that the present approach gives more accurate results than previously published approaches using the Ritz method combined with various domain composition techniques. Finally, the present approach is applied to investigate the effects of location, length and orientation of side cracks on the free vibration frequencies and mode shapes of simply supported and completely free square plates with side cracks, including cracks which are not along a symmetry axis, are skewed. Most of the results shown are novel.