The present work applies meta-heuristic multi-objective particle swarm optimization (MOPSO) and non-dominated sorting genetic algorithm (NSGA)-II algorithms to predict optimal shape and material distribution of two-directional functionally graded tapered (2DFG-T) microbeams. The cross-section of the microbeam changes linearly along its length in both height and width directions. The rates of these changes are taken as the geometric design variables. Rather than the simple power law, two different trigonometric functions are introduced to provide various effective and flexible 2D- distributions of metal and ceramic phases. Using Hamilton’s principle, the mathematical model is derived based on the third order shear deformation theory (TSDT) and modified couple stress theory (MCST). Developing a computationally efficient and time saving semi-analytical approach, the governing set of variable-coefficients partial differential equations is solved for the frequency and critical buckling under different boundary conditions. Several constrained single- and multi-objective optimization problems are formulated and solved to determine optimal material and geometry parameters of 2DFG-T microbeams to achieve maximization of fundamental frequency and/or critical buckling load while minimizing the structural mass. Various optimization results demonstrate that including shape parameters in addition to material distribution parameters as design variables results in more economical and flexible profiles that is critical for the manufacture of 2DFG-T microbeams. The following model can be exploited in the optimum design of FG microscale nanobeams including their shapes and material distribution.