Searches for rare Bs0 and B0 decays into four muons are performed using proton-proton collision data recorded by the LHCb experiment, corresponding to an integrated luminosity of 9 fb−1. Direct decays and decays via light scalar and J/ψ resonances are considered. No evidence for the six decays searched for is found and upper limits at the 95% confidence level on their branching fractions ranging between 1.8×10−10 and 2.6×10−9 are set.