The purpose of this article is threefold. In 1971, H. L. Royden introduced the Kobayashi-Royden pseudometrics on (nonsingular) complex manifolds and gave a criterion for the Kobayashi hyperbolicity of (nonsingular) complex manifolds which plays an essentially important role in Hyperbolic Complex Geometry. Perhaps, since there are some unresolved technical problems, the above theorem of Royden is not yet completely cleared up for singular complex spaces. Thus, the first is to prove Royden’s theorem for singular complex spaces. The second is to give a criteria of the hyperbolicity modulo a closed subset of singular complex spaces from the viewpoint of comparing the Kobayashi k-differential pseudometrics on these spaces and the Landau property of the ones. The third is to investigate the hyperbolicity modulo a closed subset of domains in a singular complex space from the condition on localization at their boundary points.