Dynamical systems need to obey to the conflicting requirements of being light-weight, self-supporting and having good noise and vibration performance. Sandwich panels have shown their potential to achieve these properties. The design of the sandwich panel cores is, however, a cumbersome task due to the multi-physical nature of the problem and the necessary trade-off between the requirements. Optimization routines have played a crucial role in this regard to automate the design of the sandwich panel cores. While parametric and size optimization of predefined cores are widely covered in the literature, studies on systematic topological design are scarce. Recently, the authors presented a topology optimization framework for the design of sandwich panels while assuming infinite periodicity [1]. This, however, can lead to unaccounted phenomena such as edge effects, when going towards a finite structure with particular boundary conditions. This work proposes, therefore, a topology optimization for 2D finite vibro-acoustic sandwich panels with as objective the sound transmission minimization, computed with the Rayleigh integral [2]. Constraints are put on both the volume and static stiffness to achieve the light-weight and self-supporting criteria. A systematic design analysis is executed with the proposed framework to investigate novel core topologies for different boundary conditions. Also the impact of whether or not imposing periodicity in the core is investigated. From the analysis, novel sandwich panel cores are presented which obey with the conflicting requirements of being light-weight, self-supporting and obtaining an increased STL performance. [1] Cool, V., et al. (2024). Vibroacoustic topology optimization for sound transmission minimization through sandwich structures. J. Sound Vibr., 568, 117959. [2] F. Fahy, P. Gardonio, Sound and structural vibration: radiation, transmission and response, 2nd edition, Elsevier, UK, 2007.