This paper is devoted to the study of the stability of the Lagrangian point L4 in the spatial restricted three-body problem and to the possibility of inclined Trojan-like objects in exoplanetary systems (single and binary star systems). The stability is investigated by numerical methods, computing stability maps in different parameter planes. In the case of circular motion of the primary bodies, it is shown that there are stable orbits up to an inclination i = 61° of the test particle. At moderate inclinations (∼10° to ∼50°), we find that the stability limit in the mass ratio of the primaries extends well beyond the linear stability value of 0.0385 – with stable orbits existing even for extreme mass ratios of 0.048. In the case of elliptic motion of the primaries, the stable region in the mass ratio–eccentricity plane shrinks as the inclination increases, with no stable orbits being found for inclinations in excess of i = 61°. Both in the circular and elliptic cases, the structure of the stability regions is closely connected with secondary resonances between the librational frequencies. As an application, the results are applied to 35 known exoplanetary systems showing which of them may possess Trojan-like objects in inclined orbits.