In　this　paper,　we　investigate　the　free　sets　and　the　free　subsemimodules　in　a　semimodule　over　a　commutative　semiring　S.　First,　we　discuss　some　properties　of　the　free　sets　and　give　a　sufficient　condition　for　a　nonempty　finite　set　to　be　free　in　a　finitely　generated　free　S-semimodule　and　obtain　a　relation　between　free　set　and　linear　independent　set　in　an　S-semimodule.　Then　we　consider　the　free　subsemimodules　and　prove　that　the　rank　of　any　free　subsemimodule　of　a　finitely　generated　S-semimodule　M　does　not　exceed　that　of　M.　Also,　we　give　some　equivalent　descriptions　for　a　commutative　semiring　S　to　have　the　property　that　all　nonzero　subsemimodules　of　any　finitely　generated　free　S-semimodule　are　free.　Partial　results　obtained　in　the　paper　develop　and　generalize　the　corresponding　results　for　modules　over　rings　and　linear　spaces　over　fields.　©　2016　Elsevier　Inc.　All　rights　reserved.