Abstract We use the Mindlin plate theory and the finite element method to delineate the effect of fixing points on a transverse normal to the mid-surface of the plate on the localization of buckling modes in clamped–clamped rectangular plates made of linear elastic, homogeneous and either isotropic (monolithic) or orthotropic (fiber-reinforced composite) materials. The in-plane loads considered on the bounding edges are: (i) normal tractions on the length, (ii) normal tractions on the width, (iii) equal normal tractions on the length and the width (equal biaxial loading), (iv) shear (tangential), and (v) combined same normal and shear tractions on all sides. It is found that clamping points on a transverse normal passing through the mid-point of a line parallel to the short side increases the critical buckling load of plates of only low aspect ratios over that of the corresponding plates unconstrained at interior points. However, for plates of all aspect ratios (thickness/length) fixing points on a transverse normal divides it into two regions with negligible transverse deflections in only one of the two regions. Only for loads (i)–(iii) the dividing line is parallel to the short side of the plate. For both thin and thick isotropic plates the slope of the dividing line is found to monotonically increase with an increase in the aspect ratio of a plate until it reaches a saturation value. A parameter based on the modal strain energy is used to quantify the degree of localization of a buckling mode. For an isotropic plate the degree of localization is found to increase with the increase in the aspect ratio for load cases (i)–(iii) but is found to be moderate for load cases (iv) and (v). For an orthotropic layer the degree of localization with an increase in the aspect ratio of the plate increases more for the 90° lamina than that for the 0° and the 45° laminae. Also, the mode localization in the (45°, −45°) laminate is stronger than that in the (0°, 90°) laminate for the five load cases. However, moderate degree of mode localization is found in symmetric and anti-symmetric cross-ply and angle-ply laminates.