Dislocations represent one of the most fascinating and fundamental concepts in materials science(1-3). Most importantly, dislocations are the main carriers of plastic deformation in crystalline materials(4-6). Furthermore, they can strongly affect the local electronic and optical properties of semiconductors and ionic crystals(7,8). In materials with small dimensions, they experience extensive image forces, which attract them to the surface to release strain energy(9). However, in layered crystals such as graphite, dislocation movement is mainly restricted to the basal plane. Thus, the dislocations cannot escape, enabling their confinement in crystals as thin as only two monolayers. To explore the nature of dislocations under such extreme boundary conditions, the material of choice is bilayer graphene, the thinnest possible quasi-two-dimensional crystal in which such linear defects can be confined. Homogeneous and robust graphene membranes derived from high-quality epitaxial graphene on silicon carbide(10) provide an ideal platform for their investigation. Here we report the direct observation of basal-plane dislocations in freestanding bilayer graphene using transmission electron microscopy and their detailed investigation by diffraction contrast analysis and atomistic simulations. Our investigation reveals two striking size effects. First, the absence of stacking-fault energy, a unique property of bilayer graphene, leads to a characteristic dislocation pattern that corresponds to an alternating AB <-> AC change of the stacking order. Second, our experiments in combination with atomistic simulations reveal a pronounced buckling of the bilayer graphene membrane that results directly from accommodation of strain. In fact, the buckling changes the strain state of the bilayer graphene and is of key importance for its electronic properties(11-14). Our findings will contribute to the understanding of dislocations and of their role in the structural, mechanical and electronic properties of bilayer and few-layer graphene.