What citizenship/place of birth would a child born on an airplane have?

The child’s place of birth will depend on the administrative practice of the authorities issuing documents. If the child acquires more than one nationality, it could have different places of birth in different documents.

On the question of the child’s nationality, the other answers are wrong. The child’s nationality depends on several factors, and the child may acquire multiple nationalities:

• the child may acquire nationality from the parents. For example, most children born to US citizen parents will be US citizens regardless of the place of birth. Most countries have similar provisions. If the parents have different nationalities or multiple nationalities, the child may acquire more than one nationality from the parents. In some cases, a parent’s ability to pass citizenship to a child is different depending on whether the birth takes place in the parent’s country of citizenship. If so, and the aircraft is in that country’s airspace at the time of the birth, it will be up to the country’s legal system to establish whether birth in the country’s airspace is to be treated the same as birth in its territory.

• the child may acquire the nationality of the aircraft’s flag country. This will be the case only if the nationality law of the flag country so provides. The Convention on the Reduction of Statelessness, mentioned in other answers, requires participating countries to grant nationality in these circumstances only if the child has no other way to acquire at least one nationality. Nothing stops a country from granting citizenship to such children even if they do acquire another nationality, however.

• the child may acquire the nationality of the country in whose airspace the birth occurs. Again, this will be the case only if the nationality law of the country so provides. If the country does not grant citizenship based on birth in its territory, then this is unlikely to be the case. If the country does grant citizenship to those born in its territory, there will be the additional question of whether airspace is considered to be part of the country’s territory for that purpose.

In short, without knowing the the parents’ country or countries of nationality, the country’s airspace in which the birth occurred, if any, and the flag country of the aircraft, it’s impossible to say which of those countries’ nationalities the child will have.

Conveniently as ever, Wikipedia has a page on Birth aboard aircraft and ships.

The law on the subject, despite the provisions of Article 3 the
1961 Convention on the Reduction of Statelessness, is complex, because
various states apply differing principles of nationality, namely jus
soli and jus sanguinis, to varying degrees and with varying
qualifications.

In general:

Under the 1944 Convention on International Civil Aviation, articles
17–21, all aircraft have the nationality of the state in which they
are registered, and may not have multiple nationalities. For births,
the law of the aircraft’s nationality is applicable, and for births
that occur in flight while the aircraft is not within the territory of
any state, it is the only applicable law. However, if the aircraft is
in or flying over the territory of another state, that state may also
have concurrent jurisdiction, and the locus in quo principle may apply
to the exact position of the aircraft when the birth occurred.

There are still very few Member States that are party to the 1961
Convention. Furthermore, conflicts of laws still exist, in particular
between the laws of North and South American states, which typically
adhere to the jus soli principle, and the laws of European states,
which usually adhere to the jus sanguinis principle.

The U.S. exception:

U.S. law holds that natural persons born on foreign ships docked at
U.S. ports or born within the limit of U.S. territorial waters are
U.S. citizens. An important exception to this rule is that children
born to people who (in line with the Fourteenth Amendment to the
United States Constitution) are not “subject to the jurisdiction” of
the United States (e.g. diplomats accredited with the United States
Department of State) are not automatically U.S. citizens. Despite a
common misconception to the contrary, birth on board a U.S.-flagged
ship, airliner, or military vessel outside of the 12-nautical mile
(22.2 km/ 13-13/16 st. mi.) limit is not considered to be a birth on
U.S. territory, and the principle of jus soli thus does not apply.

According to the 1961 Convention on the Reduction of Statelessness, Article 3:

For the purpose of assigning nationality, birth on a ship or aircraft
shall amount to birth in the territory of the State that gives its
flag to that ship or aircraft.

However, only about 40 nations (not including the USA) have ratified this convention – and what territory a birth has occurred on is not necessarily what determines citizenship.

So in reality, it’s a huge mess between:

• what country the plane is registered in
• what country’s territory the plane was flying over at the moment of birth
• the nationality of the mother and father
• whether these countries’ laws claim jurisdiction over in-air births
• whether these countries’ laws are based on the jus soli or the jus sanguinis principle
• whether these countries’ laws allow dual nationalities

In theory, you may have cases where none of the countries involved would grant the baby citizenship, or where two (or even three) would grant it automatic and exclusive citizenship.

In practice, I suspect that in almost all cases at least one of them would grant citizenship, and it’s up to the mother which one she applies for – and up to the bureaucrats how difficult that is.

A separate question is what the “place of birth” on the birth certificate will say – I suspect that’s up to the discretion of said bureaucrats, since it will have little importance in most cases.

Update:
I recently found an article that has a real-world example. It pretty much agrees with what I wrote, namely the person portrayed got her mother’s UK citizenship with a passport note saying “Holder born on an aeroplane 10 miles south of Mayfield, Sussex.”, which later had to be changed to “born at sea” to comply with an EU directive. US State Department rules, on the other hand, would list the place of birth as “in the air”.

In principle the law of the native country of the carrier decides. So the answer first really depends on the airline. Then there are complicating factors. Some countries dictate their nationality to offspring of their nationals no matter where they are born. Other countries say that once born on native soil you are by definition their citizen.

So the main answer is that is depends on the the local law of the carrier.