In a bowl fit with a mixer, whip margarine/butter, and powdered confectioners' sugar at medium speed for about 7 minutes. Whipping it this long will make the cookies light and tender.
Meanwhile, in a medium bowl, whisk together flour, cornstarch, and salt; set aside.
Carefully add 1/2 cup of flour mixture at a time to mixer. After each addition, once flour is partially absorbed, turn mixer to medium to fully incorporate flour into butter. Turn back to low and repeat with remaining flour mixture.
Cover bowl and refrigerate for at least 30 minutes.
Preheat oven to 300 degrees F. Line a baking sheet with parchment paper.
With hands lightly dusted with additional cornstarch, roll dough into 1-in. balls or about 1/2 tablespoon cookie dough.
Place 1 in. apart on ungreased baking sheets. Press lightly with a floured fork. Top with nonpareils or cherry halves.
Bake 20-22 minutes. Allow to cool then move to cooling rack. Store in air-tight container.