This paper provides a detailed comparison in a solids mechanics context of adaptive mesh refinement methods for all-quadrilateral and all-hexahedral meshes. The adaptive multigrid Local Defect Correction method and the well-known hierarchical h-adaptive refinement techniques are placed into a generic algorithmic setting for an objective numerical comparison. Such a comparison is of great interest as local multigrid AMR approaches are from now rarely employed to adaptively solve implicit systems in solid mechanics. However they present various interesting features mainly related to their intrinsic idea of partitioning the degrees of freedom on different mesh levels. For this study, we rely on a fully-automatic mesh refinement algorithm providing the desired refined mesh directly from the user-prescribed accuracy. The refinement process is driven by an a posteriori error estimator combined to mesh optimality criteria. In this study, the most efficient strategy based on mesh optimality criterion and refinement ratio is identified for all-quadrilateral and all-hexahedral finite elements meshes. The quality of refined meshes is finally appreciated in term of number of nodes but also through the verification of final solution’s accuracy. A special attention is devoted to the fulfillment of local precisions which are of great importance from an engineering point of view. Numerical 2D and 3D experiments of different complexities revealing local phenomena enable to highlight the essential features of the considered mesh refinement methods within an elastostatic framework. This study points out the great potentialities of locally adaptive multigrid method, which clearly appears to be the most powerful strategy in terms of standard metrics of efficiency (dimension of systems to be solved, storage requirements, CPU time).