Stable motions around the triangular libration points (TLP) in the Earth-Moon system perturbed by Sun are described. Two special quasi-periodic orbits around each TLP (called dynamical substitutes in this paper) are given. They were previously described in the planar Sun-Earth-Moon system by Kamel and are numerically obtained in the spatial Sun-Earth-Moon system. Linearized motions around the dynamical substitute are described in a semi-analytical way. The size of the stable region around the substitute is described via a numerical approach, along with the resonance mechanisms that determine the boundary of the stable region. Possible temporary capture from the near Earth objects population is discussed. An observation algorithm to search permanent or temporarily captured objects in this region is given. Some potential applications of the TLPs of the Earth-Moon system in space missions are discussed.