This paper explores the rich dynamics of quasi-satellite orbits with out-of-plane motions (spatial QSOs) in the Earth-Moon and Mars-Phobos circular restricted three-body problems. The first part of the paper reveals families of spatial periodic QSOs via bifurcation analyses, and presents orbital characteristics of each family. In each of the Earth-Moon and Mars-Phobos systems, we find weakly unstable spatial families, which could be useful for observation and exploration of high-latitude regions of a celestial body. The second part of the paper discusses three applications of the obtained spatial unstable periodic QSOs for space mission trajectories. The first application preliminarily investigates a ballistic landing option on the surface of the Phobos via unstable manifolds emanating from weakly unstable spatial periodic QSOs. The second application explores the relationship between stability regions of spatial, long-term stable, quasi-periodic QSOs and phase-space structures of invariant manifolds emanating from spatial unstable periodic QSOs. The third application proposes a method of designing nearly-ballistic, two-impulse transfers from a low Earth orbit to a spatial, long-term stable, quasi-periodic QSO around the Moon in the bicircular restricted four-body problem including solar perturbation.