Summary: The spin-orbit interactions (SOI) for the single and double ring-shaped oscillator potentials are studied as an energy correction to the Schrödinger equation. We find that the degeneracy for the energy levels with angular quantum number \(m = 0\) keeps invariant in the case of the SOI. The degeneracy is still 2 for single ring-shaped potential and 4 for double ring-shaped potential. However, for the energy levels with angular quantum number \(m \neq 0\) the degeneracy is reduced from original 4 for the single ring-shaped potential and 8 for the double ring-shaped potential to 2. That is, their energy levels in the case of the SOI are split to 2 (single) and 4 (double) sublevels. There exists an accidental degeneracy for the cases \(|m|=2,3,4,\dots\). We note that around the critical value \(b_0\), the energy levels are reversed. We also discuss some special cases for \(\eta=2,3,4,5,6,\dots\), and the \(b=0\), \(c>0\). It should be pointed out that the parameter \(b_0\) is relevant for the angular part parameter \(b\) in the single and double ring-shaped potentials and it makes the energy levels changed from positive to negative, but the parameter \(c\) corresponds to the angular part parameter in double ring-shaped potential and the \(\eta\) is related to it. This model can be useful for investigations of axial symmetric subjects like the ring-shaped molecules or related problems and may also be easily extended to a many-electron theory.