Recently, metamaterials, sandwich panels, and a combination of both have shown potential for creating lightweight, load-bearing structures with good noise and vibration suppression properties. However, designing these structures is difficult due to the complex vibroacoustic innate physics and the need to balance conflicting requirements. Structural optimization methods can help address this multi-functional, multi-physical design challenge. While much research has been conducted on optimizing the materials and sizes of plates and sandwich cores, the systematic topological design of fully coupled vibroacoustic cores has not yet been explored. To address this gap, this work presents a topology optimization framework for the vibroacoustic design of sandwich structure cores, with the goal of minimizing sound transmission while constraining volume and structural stiffness. The framework is used to conduct a systematic design analysis, focusing on the dynamic behavior of the optimized structures. The versatility of the methodology is demonstrated by analyzing different targeted frequency ranges, different angles of incidence and the trade-off between the acoustic and structural performance. The resulting designs are lightweight, load-bearing, and achieve high sound transmission loss performance, exceeding the mass law by 15–40 dB in targeted frequency ranges of 500Hz in the interval between 1000Hz and 3000Hz.