This　paper　focuses　on　the　Rayleigh-Taylor　instability　in　the　system　of　equations　of　the　two-dimensional　nonhomogeneous　incompressible　elasticity　fluid　in　a　horizontal　periodic　domain　with　infinite　height.　First,　we　use　variational　method　to　construct　(linear)　unstable　solutions　for　the　linearized　elastic　Rayleigh-Taylor　problem.　Then,　motivated　by　the　Grenier＇s　idea　in　[1.0],　we　further　construct　approximate　solutions　with　higher-order　growing　modes　to　the　elastic　Rayleigh-Taylor　problem　due　to　the　absence　of　viscosity　in　the　system,　and　derive　the　error　estimates　between　both　the　approximate　solutions　and　nonlinear　solutions　of　the　elastic　Rayleigh-Taylor　problem.　Finally,　we　prove　the　existence　of　escape　points　based　on　the　bootstrap　instability　method　of　Hwang-Guo　in　[25],　and　thus　obtain　the　nonlinear　Rayleigh-　Taylor　instability　result,　which　presents　that　the　Rayleigh-Taylor　instability　can　occur　in　elasticity　fluids　with　small　elasticity　coefficient.　(C)　2022　Elsevier　Inc.　All　rights　reserved.