The types of sugar I have used for these recipes are superfine or fine granulated sugar, powdered sugar, turbinado sugar, and light or dark brown sugar. My preference is cane sugar.
Refined superfine granulated sugar results in the finest texture and most neutral flavor in many baked goods such as cakes and cookies, adding a sweetness that balances and accentuates the other ingredients. Fine granulated cane sugar can be substituted for superfine sugar; you can also process fine granulated sugar in a food processor when superfine is desired. In most cookies, for example, superfine sugar will give a finer crumb with less cracking.