The aim of this work is to review and also explore even further the escape properties of orbits in a dynamical system of a two-dimensional perturbed harmonic oscillator, which is a characteristic example of open Hamiltonian systems. In particular, we conduct a thorough numerical investigation distinguishing between trapped (ordered and chaotic) and escaping orbits, considering only unbounded motion for several energy levels... It is of particular interest, to locate the basins of escape towards the different escape channels and connect them with the corresponding escape periods of the orbits. We split our examination into three different cases depending on the function of the perturbation term which determines the number of escape channels on the physical space. In every case, we computed extensive samples of orbits in both the physical and the phase space by integrating numerically the equations of motion as well as the variational equations. In an attempt to determine the regular or chaotic nature of trapped motion, we applied the SALI method as a chaos detector. It was found, that in all studied cases regions of trapped orbits coexist with several basins of escape. It was also observed, that for energy levels very close to the escape value the escape times of orbits are large, while for values of energy much higher than the escape energy the vast majority of orbits escape very quickly or even immediately to infinity. The larger escape periods have been measured for orbits with initial conditions in the boundaries of the escape basins and also in the vicinity of the fractal structure. Most of the current outcomes have been compared with previous related work. We hope that our results will be useful for a further understanding of the escape mechanism of orbits in open Hamiltonian systems with two degrees of freedom. (read more)