We have investigated the low-temperature magnetism of sodium superoxide (NaO$_{2}$), in which spin, orbital, and lattice degrees of freedom are closely entangled. The magnetic susceptibility shows anomalies at $T_{1}=220$ K and $T_{2}=190$ K, which correspond well to the structural phase transition temperatures, and a sudden decrease below $T_{3}=34$ K. At 4.2 K, the magnetization shows a clear stepwise anomaly around 30 T with a large hysteresis. In addition, the muon spin relaxation experiments indicate no magnetic phase transition down to $T=0.3$ K. The inelastic neutron scattering spectrum exhibits magnetic excitation with a finite energy gap. These results confirm that the ground state of NaO$_{2}$ is a spin-singlet state. To understand this ground state in NaO$_{2}$, we performed Raman scattering experiments. All the Raman-active libration modes expected for the marcasite phase below $T_{2}$ are observed. Furthermore, we find that several new peaks appear below $T_{3}$. This directly evidences the low crystal symmetry, namely, the presence of the phase transition at $T_{3}$. We conclude the singlet-ground state of NaO$_{2}$ due to the spin-Peierls instability.
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