We study the paths of motion of harmonic oscillators on three dimensional space. The general description of this motion is well studied and presented in upper division mechanics textbooks and it is well-known that the so called commensurate condition for angular velocities leads to Lissajous curves that are paths in 2D or 3D space on which the oscillator repeats its motion. We take a closer look at Lissajous curves and highlight that these repetitive paths can be classified into two groups: One group exhibits repetitive motions that are pendulum-like (we shall refer to this group as P-type) where the oscillations have two turning points at which the oscillator momentarily comes to a complete stop and reverses direction (just like an oscillating pendulum). The second group involves repetitive motions that are ”loop-like” (which we shall refer to this group as L-type) for which there are no turning points and the oscillator travels on closed loops without changing direction. Specifically, in the absence of damping, we investigate perturbation of phase angles on the path of oscillations and whether a small perturbation can result in formation of bands around the path of motion, while the shape of the path is preserved. We then investigate the effects of damping (linear and nonlinear) on these paths.