The present paper is devoted to a comparative study of mesh-step AMR (adaptive mesh refinement) techniques, generally better adapted for industrial applications. The well-known h-adaptive methods (both remeshing and hierarchical) and the multigrid Local Defect Correction approach are compared in the context of elastostatics for linear quadrangular finite elements. The studied AMR approaches are combined with the recovery-based a posteriori Zienkiewicz and Zhu error estimator. The detection of regions requiring refinement is carried out based on different considerations about the maximal permissible element-wise error in an optimal mesh (so-called mesh optimality criterion). Various refinement strategies related to the use of different refinement ratios (uniform or adjusted) are also considered. The quality of a refined mesh is finally appreciated by the verification of both global and local accuracy. So far, the local accuracy is quite never checked in the literature while it is of great importance from an engineering point of view. Numerical examples (academic and industrial) enable to compare the efficiency of each AMR method, especially in terms of CPU time and memory space to reach the given error thresholds. The best numerical choices in term of mesh optimality criterion and refinement strategy are also discussed.