The improved supercell plane wave expansion method is applied to theoretically study the propagation of flexural waves in a ternary locally resonant phononic crystal thin plate with a point defect and linear defects. The thin concrete plate composed of a square array of steel cylinders hemmed around by rubber is considered here. Absolute band gaps in low frequency are obtained. For the point defect, the defect mode is localized around the defect, and the magnitude of the resonant defect mode is strongly dependent on the defect filling fraction, mass density and Young’s modulus of the defect cylinder. For the straight linear defects, several resonant linear defect bands appear inside the absolute band gap. And the displacement distributions show that the flexural waves could well propagate along the linear defects.