An anchor sheet subfloor that includes a laminate having an upper layer of relatively thin flexible rigid sheet material and a bottom layer of a relatively resilient cushioning material. The upper sheet layer can be formed of a plastic or polymer material. In one arrangement, the sheet can be cut and fit within the boundaries of a room and the sheet has sufficient rigidity and mass to remain without distortion or buckling within the room by free floating on the existing floor without substantial attachment to the floor. It can be possible for a sheet to be cut and fit on site to fit the contours of a room to form by itself or in combination with other anchor sheets a free floating smooth subfloor on which can be overlaid decorative covering pieces.