In this article, we present a numerical study on stabilization and eigenmodes of the so-called skyrmion chiral spin texture in nanometric dots. The first aim of this study is to identify the appropriate multilayer in a set of Pt/Co/MgO structures with different Co thicknesses that have been previously experimentally characterized. Stabilization occurs if the energy favoring skyrmions is greater than the geometric mean of the exchange and anisotropy energies. Both the energy favoring skyrmions and the anisotropy contribution depend on the Co thickness. The appropriate multilayer is obtained for a specific Co thickness. MuMax simulations are used to calculate the precise static magnetization configuration for the experimental parameters, allowing us select the appropriate structure. Moreover, in view of experimental study of skyrmion dynamics by means of Brillouin light scattering, the eigenfrequency, eigenmode profile, and spectral density are calculated for different dot sizes. Finally, the optimal dot size that allows for a feasible experiment is obtained.