The influence of core-hole delocalization for x-ray photoelectron, x-ray absorption, and x-ray emission spectrum calculations is investigated in detail using approaches including response theory, transition-potential methods, and ground state schemes. The question of a localized/delocalized vacancy is relevant for systems with symmetrically equivalent atoms, as well as near-degeneracies that can distribute the core orbitals over several atoms. We show that the issues relating to core-hole delocalization are present for calculations considering explicit core-hole states, e.g., when using a core-excited or core-ionized reference state or for fractional occupation numbers. As electron correlation eventually alleviates the issues, but even when using coupled-cluster single-double and perturbative triple, there is a notable discrepancy between core-ionization energies obtained with localized and delocalized core-holes (0.5 eV for the carbon K-edge). Within density functional theory, the discrepancy correlates with the exchange interaction involving the core orbitals of the same spin symmetry as the delocalized core-hole. The use of a localized core-hole allows for a reasonably good inclusion of relaxation at a lower level of theory, whereas the proper symmetry solution involving a delocalized core-hole requires higher levels of theory to account for the correlation effects involved in orbital relaxation. For linear response methods, we further show that if x-ray absorption spectra are modeled by considering symmetry-unique sets of atoms, care has to be taken such that there are no delocalizations of the core orbitals, which would otherwise introduce shifts in absolute energies and relative features.