What can go in
Almost every honest answer here is conditional, and the conditions are the useful part. Foil is fine weighted under food and a fire risk loose in a drawer — a yes/no column cannot carry that difference, so we do not use one.
Your appliance manual governs over anything here. Where it prohibits something, that prohibition stands whatever the physics permits.
Foil and film — Depends on your model
Aluminium foil
One manufacturer prohibits it and says why. Another puts it in its own recipe. Your manual decides, and this page explains what each of them means.
Paper and card — Safe only in specific conditions
Baking parchment
Two manufacturers now prohibit it outright and one gives the mechanism. Sprint 2 recorded every manufacturer as silent; that is no longer true, and the verdict has hardened.
Paper and card — Best avoided
Kitchen paper and paper plates
Do not. Loose, dry, unweighted paper in a chamber with a fan and an element is the clearest fire hazard in this section.
Silicone and plastics — Safe only in specific conditions
Silicone liners and moulds
Heavy enough not to be lifted, and rated above the temperatures these appliances reach. The airflow question is the one that remains.
Cookware and bakeware — Safe only in specific conditions
Metal tins, trays and cookware
Explicitly permitted by one manufacturer for baking, and recommended by three others whenever a batter is involved.
Cookware and bakeware — Safe only in specific conditions
Baking tins
Not optional for baking. Every chart in this catalogue that has a cake row also says to use a tin.
Cookware and bakeware — Safe only in specific conditions
Ramekins and small oven dishes
Small enough to leave airflow around them, which is what makes them the most practical cookware for a drawer machine.
Glass and ceramic — Safe only in specific conditions
Oven-safe glass
Explicitly permitted by one manufacturer for baking. Thermal shock, not heat, is the thing to manage.
Glass and ceramic — Depends on your model
Ceramic and stoneware
One manufacturer prohibits it for air frying outright. That single prohibition is why this answer depends on your model.
Food packaging — Unsafe
Cardboard and food packaging
No. Dry cardboard next to a heating element is the clearest fire risk in this section and no manufacturer permits it.
Wood and natural materials — Safe only in specific conditions
Wooden and bamboo skewers
A manufacturer recommends wooden cocktail sticks for holding light food down, which is about as clear a permission as this section gets.
Silicone and plastics — Unsafe
Plastic containers and tubs
No. Microwave-safe is not oven-safe, and an air fryer runs at temperatures that will deform or melt domestic plastics.