A building flooring system includes a plurality of polymer-based composite structural underlayment boards overlaying and secured to a building subfloor to form an underlayment layer over the building subfloor and a nonstructural finish-flooring layer overlaying and bonded to the underlayment layer. The polymer-based composite structural underlayment boards each have a polymer material core layer or a predominantly polymer material core layer that typically contains from 1% to 40% by weight filler(s) and a density between 1.6 lbs/ft.sup.3 and 25 lbs/ft.sup.3. At least the top or bottom major surface of each of the polymer-based composite structural underlayment boards is overlaid by a facer that reinforces and strengthens the polymer-based composite structural underlayment board and that is generally coextensive with and bonded to the overlaid major surface of the polymer-based composite structural underlayment board.