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Buckling of thin-walled and load-bearing elements of a structure can have devastating consequences. Hence, buckling checks are an integral part of strength analysis of structures. The buckling problem of thin rectangular plates subjected to in-plane compressive and/or shear loading is of great importance in building, bridge, aerospace, marine, and shipbuilding industries. When buckling occurs, thin plates undergo large out-of-plane deflections, which in turn results in the development of large bending stresses and eventually complete failure of the structure. This paper deals with the buckling stability assessment of uniaxially-compressed plates with different support conditions within the framework of classical plate theory. The main objective of this research is to explore some uncovered aspects of buckling stability of plates by considering the effects of support conditions, aspect ratio, and slenderness ratio, which will consequently result in efficient design of such thin-walled structures. To this end, in addition to validation of the numerical simulation, some case studies have been performed in order to gain a better understanding of different aspects of buckling stability of such thin-walled structures.