In almost every case, wine is gluten-free to 20 parts per million. However, there are a few exceptions to this rule.
If the wine in question has any added coloring or flavoring (fruit-flavored dessert wines, for example, often include added ingredients), then it might not be gluten-free you'll need to contact the manufacturer to make sure.
If you're extremely sensitive to gluten cross-contamination, you may find yourself reacting to certain wines even wines without additives. You're not imagining your symptoms there are a couple of other ways gluten can sneak into wine.
In some cases, the culprit will be the use of wheat gluten as a fining, or clarifying, agent for the wine. In other cases, the culprit will be the wheat paste used to seal the wooden wine casks used to age the wine.
Both of these practices add a tiny amount of gluten to the finished wine perhaps in the range of 1 to 2 parts per million. Most people won't notice symptoms from this, but the most sensitive will.
If you do notice symptoms from wine, you can consider looking for varieties that are aged in stainless steel casks. You may also need to contact individual vineyards to see what fining agents they use.
If you find a wine you like (and that doesn't cause gluten symptoms), buy a case and stick with it.