Vertical sloshing is the liquid surface motion in a container forced to oscillate in the vertical direction. The present paper concerns the vertical sloshing in various equilateral-polygonal-section containers such as octagonal-, heptagonal-, hexagonal-, pentagonal-, square-and triangular-section containers together with a circular-section container, in order to generalise their sloshing modes. As a result, the authors classify the sloshing modes on the basis of the conventional circular-section-container sloshing modes.It is revealed that this modal classification has some advantages over that based on the conventional square-section-container sloshing modes. Furthermore, the stability diagrams for all the equilateral-polygonal-section containers are investigated by both experiments and computations. The present computation is based on a discrete singularity method. The proposed modal classification is useful to predict the eigen frequencies. Specifically speaking, it is found that the equivalent diameter d e1 based on the hydraulic mean depth is the most adequate as a characteristic length scale to classify all the sloshing modes. The authors show a unified formula to predict the eigen frequencies, using d e1 together with the proposed modal classification.