White vinegar or just plain vinegar are typically distilled, and, if so, are gluten-free. Distilled vinegar can be distilled from wheat, corn, potatoes, beets, wood, apples and many other things. Most in the USA are not made from wheat, but are instead made from corn, potatoes or wood, which are all safe (Heinz white vinegar is distilled from corn). Distilled vinegars that are made from wheat are probably gluten-free because of the distillation process described in Frederik Willem Janssens article on this site.
Distilled vinegar made from wood are gluten-free. Wood-based vinegar is often the vinegar used in processed foods.
Malted vinegars are usually not gluten-free.
Red and white wine and balsamic vinegars are gluten-free.