A self-tapping fastener, which has been especially constructed for use in connection with the insertion and installation thereof into stucco, gypsum, wallboard, or other hard board substrates, wherein the forward end portion of the shank portion of the self-tapping fastener is provided with a pair of diametrically opposed flat sections which effectively form at least one pair of diametrically opposed tapping threads so as to effectively perform the thread tapping process despite exhibiting some abrasion of the forwardmost threads. In addition, the flat sections also effectively define recessed pockets for accommodating any dust, particles, debris, or the like, which will be generated during the thread tapping process.