This　paper　focuses　on　the　Rayleigh–Taylor　instability　in　the　two-dimensional　system　of　equations　of　nonhomogeneous　incompressible　viscous　fluids　with　capillarity　effects　in　a　horizontal　periodic　domain　with　infinite　height.　First,　we　use　the　modified　variational　method　to　construct　(linear)　unstable　solutions　for　the　linearized　capillary　Rayleigh–Taylor　problem.　Then,　motivated　by　the　Grenier’s　idea　in　(Grenier　in　Commun.　Pure　Appl.　Math.　53(9):1067–1091,　2000),　we　further　construct　approximate　solutions　with　higher-order　growing　modes　to　the　capillary　Rayleigh–Taylor　problem　and　derive　the　error　estimates　between　both　the　approximate　solutions　and　nonlinear　solutions　of　the　capillary　Rayleigh–Taylor　problem.　Finally,　we　prove　the　existence　of　escape　points　based　on　the　bootstrap　instability　method　of　Hwang–Guo　in　(Hwang　and　Guo　in　Arch.　Ration.　Mech.　Anal.　167(3):235–253,　2003),　and　thus　obtain　the　nonlinear　Rayleigh–Taylor　instability　result.　Our　instability　result　presents　that　the　Rayleigh–Taylor　instability　can　occur　in　the　fluids　with　capillarity　effects　for　any　capillary　coefficient　κ＞　0　if　the　critical　capillary　coefficient　is　infinite.　In　particular,　it　improves　the　previous　Zhang’s　result　in　(Zhang　in　J.　Math.　Fluid　Mech.　24(3):70–23,　2022)　with　the　assumption　of　smallness　of　the　capillary　coefficient.　©　2023,　Springer　Nature　Switzerland　AG.