We experimentally demonstrate storage and on-demand release of phase-sensitive, photon-number superposition states of the form α|0⟩+βe^{iθ}|1⟩ for an optical quantized oscillator mode. For this purpose, we newly developed a phase-probing mechanism compatible with a storage system composed of two concatenated optical cavities, which was previously employed for storage of phase-insensitive single-photon states [Phys. Rev. X 3, 041028 (2013)PRXHAE2160-330810.1103/PhysRevX.3.041028]. This is the first demonstration of all-optically storing highly nonclassical and phase-sensitive quantum states of light. The strong nonclassicality of the states after storage becomes manifest as a negative region in the corresponding Wigner function shifted away from the origin in phase space. This negativity is otherwise, without the phase information of the memory system, unobtainable. While our scheme includes the possibility of optical storage, on-demand release and synchronization of arbitrary single-rail qubit states, it is not limited to such states. In fact, our technique is extendible to more general phase-sensitive states such as multiphoton superposition or entangled states, and thus it represents a significant step toward advanced optical quantum information processing, where highly nonclassical states are utilized as resources.