A rail damper mounting apparatus including a clipping device adapted to clip onto a rail, and a mounting device adapted to allow an oscillator to secure thereto. The rail damper mounting apparatus forms at least three clipping points with a rail when the rail damper mounting device is installed to the rail. A rail damping system containing one or more such rail damper mounting apparatus is also disclosed. With at least three clipping points/areas in the mount apparatus, it provides a strong clipping force compressing the wedge-liked rail foot by a very stiff clip.