In nature there are various regular and irregular lattice and honeycomb structures, which have been evolved over the course of millions of years of ongoing evolution and mostly serve different functions. The frequently irregular structures of diatoms for example are characterized by high strength at low mass. At the same time, a positive influence of the structural irregularity on the vibrational behavior of the structures is assumed. Within biomimetic approaches biological structures serve as a source of inspiration for different technical applications. In this work a magnet underframes of particle accelerators is considered which is to be optimizied towards a high first eigenfrequency and stiffness and a low mass. Following the calculation of the reference structure, the reference structure is replaced by irregular and regular 3D lattice structures based on biomimetic algorithms. Parameterized constructions and optimization calculations using the evolution strategy reveal the best possible structures, which represent compromises between a high eigenfrequency and stiffness and a low mass. The results show that irregular structures have, on average, a higher first eigenfrequency and stiffness than regular lattices. In a next step, topology optimizations of the magnet support structure are performed with the goals of minimum compliance and a maximum first eigenfrequency. The results reveal necessary structural elements for reaching the defined targets at low mass. The subsequent transfer of the design suggestions from the topology optimizations to parameterized models allows a further, advanced optimization with the evolution strategy generating a large number of optimized structures. In summary, the results of both optimization calculations show a large range of structures. Best structures combine both a higher first eigenfrequency as well as a higher stiffness compared to the reference structure. The irregular lattice structures for example reveal an increase of the first eigenfrequency by a factor of 1.43 with a simultaneous stiffness gain by a factor of 1.82 compared to the reference structure. The optimized structures based on the results of topology optimizations lead to an increase in the first eigenfrequency by a factor of 1.38 while the stiffness rises by a factor of 3.17. The structural masses always remain within the permitted values. The great potential of the generated structures is shown by the fact that many structures with different eigenfrequencies can have the same stiffness. Furthermore, many developed structures with the same first eigenfrequency show a wide range in stiffness. The results indicate that a variation of the geometry, and therefore the resulting change in the mass distribution can be used to shift eigenfrequencies. However, further studies are necessary.