  A search for the non-resonant decays $B^0_s \rightarrow
\mu^{+}\mu^{-}\mu^{+}\mu^{-}$ and $B^0 \rightarrow
\mu^{+}\mu^{-}\mu^{+}\mu^{-}$ is presented. The measurement is performed using
the full Run 1 data set collected in proton-proton collisions by the LHCb
experiment at the LHC. The data correspond to integrated luminosities of $1$
and $2~\mathrm{fb}^{-1}$ collected at centre-of-mass energies of $7$ and
$8~\mathrm{TeV}$, respectively. No signal is observed and upper limits on the
branching fractions of the non-resonant decays at $95\%$ confidence level are
determined to be
  \mathcal{B}(B^0_s \rightarrow \mu^{+}\mu^{-}\mu^{+}\mu^{-}) & < 2.5 \times
10^{-9}
  \mathcal{B}(B^0 \rightarrow \mu^{+}\mu^{-}\mu^{+}\mu^{-}) & < 6.9 \times
10^{-10}.
