The differential branching fraction of the rare decay $Lambda^{0}_{b} rightarrow Lambda mu^+mu^-$ is measured as a function of $q^{2}$, the square of the dimuon invariant mass. The analysis is performed using proton-proton collision data, corresponding to an integrated luminosity of $3.0 mbox{ fb}^{-1}$, collected by the LHCb experiment. Evidence of signal is observed in the $q^2$ region below the square of the $J/psi$ mass. Integrating over $15 < q^{2} < 20 mbox{ GeV}^2/c^4$ the branching fraction is measured as $dmathcal{B}(Lambda^{0}_{b} rightarrow Lambda mu^+mu^-)/dq^2 = (1.18 ^{+ 0.09} _{-0.08} pm 0.03 pm 0.27) times 10^{-7} ( mbox{GeV}^{2}/c^{4})^{-1}$, where the uncertainties are statistical, systematic and due to the normalisation mode, $Lambda^{0}_{b} rightarrow J/psi Lambda$, respectively. In the $q^2$ intervals where the signal is observed, angular distributions are studied and the forward-backward asymmetries in the dimuon ($A^{l}_{rm FB}$) and hadron ($A^{h}_{rm FB}$) systems are measured for the first time. In the range $15 < q^2 < 20 mbox{ GeV}^2/c^4$ they are found to be $A^{l}_{rm FB} = -0.05 pm 0.09 mbox{ (stat)} pm 0.03 mbox{ (syst)}$ and $A^{h}_{rm FB} = -0.29 pm 0.07 mbox{ (stat)} pm 0.03 mbox{ (syst)}$.
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