We　study　the　nonlinear　Rayleigh–Taylor　(RT)　instability　of　an　inhomogeneous　incompressible　viscoelastic　fluid　in　a　bounded　domain.　It　is　well　known　that　there　exist　strong　solutions　of　RT　instability　in　H2-norm　in　inhomogeneous　incompressible　viscoelastic　fluids,　when　the　elasticity　coefficient　κ　is　less　than　some　threshold　κC.　In　this　paper,　we　prove　the　existence　of　classical　solutions　of　RT　instability　in　L1-norm　in　Lagrangian　coordinates　based　on　a　bootstrap　instability　method　with　finer　analysis,　if　κ<　κC.　Moreover,　we　also　get　classical　solutions　of　RT　instability　in　L1-norm　in　Eulerian　coordinates　by　further　applying　an　inverse　transformation　of　Lagrangian　coordinates.　©　2019,　The　Author(s).