This paper analyzes and discusses the main characteristics of four optimization methods used in the synthesis of planar mechanisms. Particularities of the gradient-free Nelder-Mead, gradient-based local optimization, genetic algorithm for global optimization, and the particle swarm optimization methods are discussed and applied to the synthesis of restricted and unrestricted four-bar mechanism, which is the most popular and is usually present in more complex mechanisms, with a simple and smooth desired trajectory, and a more complicated six-bar with desired points throughout the crank domain and with a complex speed requirement. This paper does not intend to establish a comparison on which one is the best, which is an ambitious and arduous task to demonstrate but rather guides the designer to a correct selection of optimization methods considering the availability of the gradient and the computational requirements of the problem. Finally, some criteria for choosing a suitable synthesis method, depending on the problem requirements, are proposed.