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 $J/\psi$ 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 $1.8\times 10-10$ and $2.6\times 10-9$ are set.