We report the computational discovery of complex, topologically charged, and spectrally stable states in three-dimensional multicomponent nonlinear wave systems of nonlinear Schrödinger type. While our computations relate to two-component atomic Bose-Einstein condensates in parabolic traps, our methods can be broadly applied to high-dimensional nonlinear systems of partial differential equations. The combination of the so-called deflation technique with a careful selection of initial guesses enables the computation of an breadth of patterns, including ones combining vortex lines, rings, stars, and vortex labyrinths. Despite their complexity, they may be dynamically robust and amenable to experimental observation, as confirmed by Bogoliubov–de Gennes spectral analysis and numerical evolution simulations.