This review paper provides a critical interpretation and analysis of almost 150 dedicated optimization research papers in the field of droop-controlled islanded microgrids. The significance of optimal microgrid allocation and operation studies comes from their importance for further deployment of renewable energy, reliable and stable autonomous operation on a larger scale, and the electrification of rural and isolated communities. Additionally, a comprehensive overview of islanded microgrids in terms of structure, type, and hierarchical control strategy was presented. Furthermore, a larger emphasis was given to the main optimization problems faced by droop-controlled islanded microgrids such as allocation, scheduling and dispatch, reconfiguration, control, and energy management systems. The main outcome of this review in relation to optimization problem components is the classification of objective functions, constraints, and decision variables into 10, 9 and 6 distinctive categories, respectively, taking into consideration the multi-criteria decision problems as well as the optimization with uncertainty problems in the classification criterion. Additionally, the optimization techniques used were investigated and identified as classical and artificial intelligence algorithms with the latter gaining popularity in recent years. Lastly, some future trends for research were put forward and explained based on the critical analysis of the selected papers.