Stacked submonolayer (SML) InAs/GaAs nanostructures, assembled by cyclic, alternating deposition of SML (<1 ML) InAs and few-monolayer GaAs using molecular beam epitaxy, have been attracting interest, owing to their unique optical and electronic properties. Recently, it has been demonstrated that a growth transition during SML deposition can lead to two types of nanostructures: 2D islands and 3D structures. The properties of SML nanostructures also make them strong candidates for spintronic and quantum information applications. However, the spin properties of SML nanostructures have not yet been investigated. In this study, the spin properties of SML nanostructures are investigated using optical spin injection and detection experiments by circularly polarized photoluminescence (CP-PL). Spins are injected into the SML nanostructures using the optical selection rules in GaAs for CP excitation, whereas the spin state in the SML nanostructures is detected by measuring the right (σ+) and left (σ−) CP intensity components of the PL. The degree of CP-PL is estimated by quantity P=[I(σ+)−I(σ−)]/[I(σ+)+I(σ−)], where I(σ±) is the luminescence intensity for the σ± component. The quantity P is a direct measure of the spin state in the SML nanostructures. Using a sample containing both 2D and 3D SML nanostructures, experimental results yield a relatively high P=6% for the 3D SML nanostructures and a relatively low P=2% for the 2D SML nanostructures. The difference may be attributed to the higher carrier confinement for 3D SML resulting in preservation of the spin state. Analytical calculations considering the spin and carrier lifetimes are also carried out to model the experimental results. These results provide insight into the fundamental spin dynamics of 2D and 3D SML nanostructures and pave the way for spintronics and quantum information applications of SML nanostructures.