The strategic choice of honeycomb cores can optimize the stability of sandwich structures while minimizing weight. This study examined three-layer composite plates with honeycomb cores, which are widely used in the structural elements of rocket and space equipment. The study aimed to compare the static stability of plates with four honeycomb cell geometries: hexagonal, auxetic, square, and triangular. The radius of the circumscribed circle for a unit cell of each honeycomb type was assumed to be identical. The honeycomb cores were fabricated via 3D printing from polylactide, and the outer layers were made of carbon-fiber-reinforced plastic. Analysis was performed using orthotropic properties of the polylactide layers, experimentally obtained. Critical loads and forms of buckling are obtained in the commercial software ANSYS. The mass of the three-layer plates increased in the following order: hexagonal, square, triangular, and auxetic. This sequence reflects the spatial arrangement of cells and the unique bending geometry of auxetic structures. Instability during bending occurred locally near the fixed edge, where cell sidewalls experienced bulging. Critical pressure for instability decreased in the following order: auxetic, hexagonal, triangular, and square honeycomb plates. In longitudinal compression, global buckling was observed in plates whose length exceeded their width. Here, critical buckling pressure followed a different order: triangular, square, auxetic, and hexagonal honeycomb plates. Instability patterns were consistent across all cell types, underscoring the impact of cell geometry on the structural performance of composite sandwich plates.