By assuming that Y(4260) either is a D_1D̅ molecular state, or has sizeable couplings with D_0 D̅^* and D_1^'D̅, we investigate several decay modes of Y(4260). Under the special kinematic configurations, triangle singularity (TS) may occur in the rescattering amplitude, which will significantly change the threshold behavior. Obvious threshold enhancements or narrow cusp structures appear quite naturally without introducing a genuine resonance. We emphasize that the radiative decay modes may be useful for studying D^(*)D̅ S-wave scattering.