Optical systems can benefit strongly from freeform surfaces; however, the choice of the right representation is not trivial, and many aspects must be considered. Many possibilities to formulate the surface equations in detail are available, but the experience with these newer representations is rather limited. Therefore, in this work, the focus is to investigate the performance of several classical descriptions as well as one extended freeform surface description in their performance in concrete design optimization tasks. There are different influencing factors characterizing the surface representations, the basic shape, the boundary function, the symmetry, a projection factor, as well as the deformation term describing higher order contributions. We discuss some possibilities and the consequences of describing and using these options with success. These surface representations were chosen to evaluate their impact on all these aspects in the design process. As criteria to distinguish the various options, the convergence over the polynomial orders, as well as the quality of the final solutions, is considered. As a result, recommendations for the right choice of freeform surface representations for practical issues in the optimization of optical systems can be given under restrictions of the benchmark assumptions.