Searches are performed for a low-mass dimuon resonance, X, produced in proton-proton collisions at a center-of-mass energy of 13 TeV, using a data sample corresponding to an integrated luminosity of 5.1 fb−1 and collected with the LHCb detector. The X bosons can either decay promptly or displaced from the proton-proton collision, where in both cases the requirements placed on the event and the assumptions made about the production mechanisms are kept as minimal as possible. The searches for promptly decaying X bosons explore the mass range from near the dimuon threshold up to 60 GeV, with nonnegligible X widths considered above 20 GeV. The searches for displaced X → μ+μ− decays consider masses up to 3 GeV. None of the searches finds evidence for a signal and 90% confidence-level exclusion limits are placed on the X → μ+μ− cross sections, each with minimal model dependence. In addition, these results are used to place world-leading constraints on GeV-scale bosons in the two-Higgs-doublet and hidden-valley scenarios. [Figure not available: see fulltext.].