Photo.net generally is a good source for such info although the thread that Holedigger linked to contains a few contradictory answers. Non-lawyers often answer this question by portraying the way they think things should be as if that were how things are.
There are at least four kinds of rights potentially implicated: Trademark, copyright, property/model releases, and contract.
Trademark: Trademarks are words or designs that identify a commercial entity and its products. Most aircraft do not have trademarks on them, but all CAF aircraft now do; sponsored air racers and aerobatic planes are often covered with them; etc. Trademarks may also appear in the background of a photo (e.g. if your photo from Oshkosh contains EAA signage). Trademark holders do not have the absolute right to prohibit the reproduction of their marks, but they can prevent the use in ways that cause consumer confusion as to the source of a product or otherwise damage the commercial value of their mark and brand. They vary in how far they will go to protect their marks.
Copyright: Arguably the paint job on an aircraft, and specific elements such as the nose art, are copyrighted by the artist who painted them or the organization that he was working for. I say "arguably" because copyright protection contains an element of originality and therefore the more closely the work imitates a previously existing work by someone else, such as an original WW2 paint scheme, the less protection it would have. In principle the artist/organization could object to the reproduction of its original artwork through a photograph, but I have not heard of it being done with vintage aircraft.
Property/model releases: If your picture includes a recognizable person who is not a pro model or someone knowingly posing for photos (or maybe even if s/he is), they may have a privacy right to control distribution of your photograph unless you have consent in the form of a written release. When an item of property is closely associated with a person such that publication of it can harm or disparage that person, that person
may be able to restrict use of the photo. There is also a theory that people have the right to profit from their property and that if you use a photo of it to make money, they
may be entitled to a piece of the action. I say "may" because the law is vague here and it is not clear when, if ever, you need a property release. But a lot of prudent photographers get them if they can.
Contract: If you were standing on private property when you took the photo, there may be conditions on your right to be there that you agreed to upon entry that limit what you can do with photos you take. Some venues, like movies and plays, prohibit photos altogether. These contracts do not give the property owner any intellectual property rights in your photos, but they may give a contract right to damages or injunctive relief if you violate their terms. Of the issues I'm listing, this is the one that most likely comes into play with airshows. Each one will differ in terms of whether it asserts any conditions.
Of the airshows/events that I'm familiar with, the EAA is known to be the most comprehensive with respect to its policies. At the link below under "Policies" there are 4 links pertaining to Airventure, 3 of which are applicable to anyone taking pictures there.
http://airventure.org/media/
Not all airshows have anything like this, and some that do may be unenforceable. For example, if I go to an airshow with cameras and there's no agreement printed on my ticket, and nobody tells me to go look for some photo policy buried on the organizer's web site, I won't necessarily feel bound by it.
August