In　this　article,　we　develop　several　first　order　fully　discrete　Galerkin　finite　element　schemes　for　the　Oldroyd　model　and　establish　the　corresponding　stability　results　for　these　numerical　schemes　with　smooth　and　nonsmooth　initial　data.　The　stable　mixed　finite　element　method　is　used　to　the　spatial　discretization,　and　the　temporal　treatments　of　the　spatial　discrete　Oldroyd　model　include　the　first　order　implicit,　semi-implicit,　implicit/explicit,　and　explicit　schemes.　The　H-2-stability　results　of　the　different　numerical　schemes　are　provided,　where　the　first-order　implicit　and　semi-implicit　schemes　are　the　H-2-unconditional　stable,　the　implicit/explicit　scheme　is　the　H-2-almost　unconditional　stable,　and　the　first　order　explicit　scheme　is　the　H-2-conditional　stable.　Finally,　some　numerical　investigations　of　the　H-2-stability　results　of　the　considered　numerical　schemes　for　the　Oldroyd　model　are　provided　to　verify　the　established　theoretical　findings.