Some sheet product dispensers disclosed herein are configured to dispense from a sheet product roll that includes a plurality of predetermined portions that are designed for providing a toilet seat cover. A line of perforations separates consecutive portions. A motor operates to drive the drive roller to cause movement of the sheet product along a dispensing pathway to position a leading predetermined portion of the sheet product into a dispensing position such that the trailing edge of the leading predetermined portion is positioned within the housing and the leading edge of the leading predetermined portion is positioned outside the dispensing chute to enable removal of the leading predetermined portion. Staged dispensing is utilized to help maintain hygiene, while still quickly providing the dispense. Perforation tear assist features are provided to aid in proper separation of the line of perforations, such as to help maintain integrity of the dispensed portion.