Plain potatoes just the spuds, fresh from the ground are gluten-free to a very low level. However, the more you do to them, the bigger the risk that they'll no longer be safe on the gluten-free diet.
Here are some common ways to prepare potatoes, and the potential gluten-related pitfalls in each:
Generally speaking, potatoes you bake at home are safe, especially if you have a completely gluten-free kitchen. If you share a kitchen, however, you should not place baking potatoes directly on the oven rack, as that may introduce gluten cross-contamination from gluten-filled rolls or pizzas.
Baked potatoes you get in restaurants may be safe (they usually are, in fact), but you'll need to check with the chef to be certain: some restaurants coat the skins in butter and flour to make them crispier, and some bake the potatoes in the oven right alongside the rolls. If in doubt, ask the restaurant staff to fix you a "clean" potato.
French fries you make at home from potato slices should be gluten-free. In addition, many brands of prepared French fries are considered gluten-free.
However, you're likely to run into problems when ordering French fries in a restaurant or a fast food outlet. The problem there isn't the fries themselves; it's the oil they're cooked in, since restaurants typically share oil between fries and wheat-coated foods such as onion rings and chicken fingers. Yes, that oil can gluten you, so avoid fries cooked in shared friers. You'll also need to steer clear of French fries with a crispy batter coating that batter typically is made of wheat.
Here's a complete rundown of safe frozen french fry brands, plus lists of chain restaurants and fast food outlets that serve safe fries:
Mashed potatoes you make from scratch are gluten-free, assuming you avoid cross-contamination in the kitchen and that you don't add any wheat flour as a thickener (once upon a time, we used to do this at my house to make mashed potato pancakes).
In addition, several brands of instant mashed potatoes are considered gluten-free to less than 20 parts per million (ppm), including Betty Crocker Potato Buds, Idaho Spuds Naturals line, and Hungry Jack Instant Mashed Potatoes.
This is a case where you'll need to check all the ingredients. Most stuffed potato recipes I've seen would be easy to make gluten-free (they didn't contain any obvious gluten-containing foods, such as bread), but most contain ingredients like bacon, cheese and instant soup mix that come in both gluten-free and gluten-filled versions. Choose your ingredients carefully if you make stuffed potatoes at home, and triple-check those ingredients before you order stuffed potatoes in a restaurant or try them at a friends' house.
Here's another case where you'll need to check all the ingredients. In addition, most restaurants fry their potato skins in shared fryers, which makes them unsafe for those of us with celiac disease and gluten sensitivity. Tread super-carefully when it comes to potato skins.
Potatoes Au Gratin
Recipes for potatoes au gratin generally are not safe: they almost always call for flour as a thickener, plus a bread crumb topping. Of course, you can make them gluten-free (this gluten-free scalloped potato recipe from About.com's Guide to Gluten-Free Cooking fits the bill nicely), but you should avoid potatoes au gratin and other potato casseroles when dining out.
Despite the name, the conventional potato bread you can buy in the store contains wheat flour, usually as the first or second ingredient. Steer clear, or make your own (here's a gluten-free and dairy-free recipe for potato scones).
Potato Flour and Potato Starch
These potato flour products appear as ingredients in numerous gluten-free recipes, and you easily can find safe sources for them. Both Bob's Red Mill and Ener-G brands offer potato starch and potato flour. Bob's tests for trace gluten to below 20ppm, or GF-20 levels, while Ener-G tests to below 5ppm, or GF-5.