A bstract Searches for rare $$ {B}_s^0 $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mi>B</mml:mi> <mml:mi>s</mml:mi> <mml:mn>0</mml:mn> </mml:msubsup> </mml:math> and B 0 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.