Since the early days of Dirac flux quantization, magnetic monopoles have been sought after as a potential corollary of quantized electric charge. As opposed to magnetic monopoles embedded into the theory of electromagnetism, Weyl crystals exhibit Berry flux monopoles in reciprocal parameter space. As a function of crystal momentum, such monopoles locate at the degeneracy point of the Weyl cone. Here, we report momentum-resolved spectroscopic signatures of Berry flux monopoles in TaAs as a paradigmatic Weyl semimetal. We have probed the orbital and spin angular momentum (OAM and SAM) of the Weyl-fermion states by angle-resolved photoemission spectroscopy at bulk-sensitive soft X-ray energies (SX-ARPES) combined with photoelectron spin detection and circular dichroism. Supported by first-principles calculations, our measurements image characteristics of a topologically non-trivial winding of the OAM at the Weyl nodes and unveil a chirality-dependent SAM of the Weyl bands. Our results experimentally visualize the non-trivial momentum-space topology in a Weyl semimetal, promising to have profound implications for the study of quantum-geometric effects in solids.