Milk, plain yoghurt, butter and almost all natural cheese are gluten free. Three things break that: blue cheese made with bread-cultured mould, processed cheese products thickened with wheat starch, and anything with cookie, biscuit or granola inclusions.
Dairy is one of the safest aisles in the shop, which is why the few exceptions catch people off guard.

Blue cheese is the famous edge case, and the honest answer is more reassuring than its reputation. Penicillium roqueforti was historically cultured on rye bread, which is where the worry comes from. Testing work on commercial blue cheeses has consistently found gluten below the 20 ppm threshold. The mould is grown on the bread, then separated, and the transfer is minimal. Most celiac organisations now treat blue cheese as safe. If you want certainty, the makers who culture on a non-grain medium say so on the packaging.
Processed cheese products are a different category from cheese. Spreads, sauces, and some pre-shredded blends use wheat starch or modified starch as a thickener or anti-caking agent. Pre-shredded cheese is usually dusted with potato or corn starch, but the label is the only way to be sure.
Yoghurt is plain milk and culture, and plain yoghurt is safe. The failures are all inclusions: granola pots, cookie-dough flavours, biscuit toppers, and malt-flavoured varieties.
One thing worth knowing that has nothing to do with labels: secondary lactose intolerance is very common at celiac diagnosis, because lactase is produced at the tips of the intestinal villi that celiac disease damages. It usually resolves as the gut heals. People often conclude they cannot tolerate dairy when what they actually cannot tolerate yet is lactose.
Point CiCi at the barcode and get this same reasoning in about two seconds, plus the carbs and whether it has been recalled. 360,000+ packaged foods are already read and waiting.
Sorted by what the answer actually is, rather than alphabetically — because the useful question in a shop is “which pile is this in.”
Malt powder is barley.
Anti-caking starch is usually potato or corn; check when it is unspecified.
Wheat starch and modified starch appear as thickeners.
Granola, cookie pieces and malt flavouring are the risks, not the yoghurt.
Single-ingredient dairy.
Milk, salt, culture, rennet.
Historically cultured on rye bread, but commercial blue cheeses consistently test below 20 ppm.
Milk and live cultures.
Cream, sugar, flavour. Cookie, brownie and malt varieties are not.
Fresh cheeses with no grain component.
Yes, in practice. The Penicillium mould was traditionally grown on rye bread, which is where the concern originates, but testing of commercial blue cheeses has repeatedly found gluten below 20 ppm and most celiac organisations treat it as safe. Producers who culture on a non-grain medium usually say so on the label.
Usually. Pre-shredded cheese is coated with an anti-caking starch, which is normally potato or corn. Where the label says only "starch" without qualifying it, it is worth checking, though on an FDA-regulated label wheat would have to be declared.
Lactase is produced at the tips of the intestinal villi, the tissue celiac disease damages, so secondary lactose intolerance is very common at diagnosis. It typically resolves as the gut heals on a gluten-free diet. This is a lactose problem rather than a gluten-in-the-dairy problem.
This site answers the question you knew to ask. The app answers the one you didn't: the barley malt in a rice cereal, the wheat starch in a gummy. Scan the barcode, or photograph the panel when the barcode isn't in any database, and CiCi shows which words on the label produced the answer.
14 days free, then $29.99 a year. Cheaper than Fig, and it does what three separate apps do. This website stays free forever either way: every answer on it, permanently, with no account.