It depends on where you are
Oats are naturally gluten-free and FDA does not classify them as a gluten grain. The problem is the supply chain: oats are commonly grown in rotation with wheat and processed on shared equipment. Uncertified oats are one of the most reliable ways to get glutened while reading labels correctly.
A cereal grain, but not one of the gluten grains. Oats are botanically distant from wheat, barley and rye, and the storage protein they carry is avenin rather than gliadin or hordein. Two separate things are therefore being argued about whenever oats come up, and conflating them is most of why the topic is confusing: whether oats are contaminated with wheat, and whether avenin itself is a problem.
Because both of those questions have real answers and they point different ways. Contamination is common and fixable. Avenin reactivity is rare and is not fixable by purity. A page that only tells you about one of them has told you half of it.
In the US, buy certified. The premium is small and it is the only thing separating a tested product from an untested one. Then, separately from contamination, know that a minority of people with celiac disease react to avenin itself — the most conservative clinical guidance declines to endorse any particular brand for that reason. If you have been scrupulous about certified oats and still feel unwell after eating them, that is a recognised pattern and worth raising with your gastroenterologist rather than assuming you got a bad box.
Oats are classed as JURISDICTION rather than safe or unsafe, which means CiCi applies the rules of wherever you are instead of the rules of wherever the packet was printed. It will tell you when a product that is fine at home would not qualify where you are travelling — which for oats is the difference between an ordinary breakfast and one that is not permitted to call itself gluten-free.
Every one of these is the same ingredient. The number of names is a large part of why gluten slips past people who read every label.
Point CiCi at the barcode on the product you're holding and get this same reasoning in about two seconds, plus the carbs and the full nutrition panel. 360,000+ packaged foods are already read and waiting.
The aisle guides below cover Oats in context — which is usually the more useful question, because an ingredient rarely appears alone.
This answer comes from CiCi's ingredient lexicon — the same rules the scanner uses, not a separate article. Where reputable sources disagree, CiCi takes the conservative position and says so.