Just curious. If a person has only one citizenship can they renounce it? How? Is it even possible and what are the consequences?
They will be stateless; not a citizen of any country.
That can also happen if a country dissolves without a successor.
Some countries accept such people as travelers, some not. You could just normally apply for a visa.
It can become quite complicated to get a passport, so typically, as a stateless person, you have to keep whatever paperwork you have that proves your identity.
Check also in the wiki for stateless for many details: https://en.wikipedia.org/wiki/Statelessness
Well, there's a great thread on reddit from a few years back on this.
Some key points:
most countries, including the US, will not allow voluntary renouncing of citizenship unless you can provide evidence you have another citizenship to fall back onto
one of the commenters renounced their Soviet citizenship in 1990 and lived in Finland for 8 years before gaining Finnish citizenship, living on an "Alien Passport".
without citizenship, you won't have a valid document for travel, which will make movement throughout the world really difficult
most jobs require proof of citizenship, or valid issued work visa. If you don't have a work visa (in a valid passport), you will struggle to get employment as well.
If a person has only one citizenship can they renounce it? How? Is it even possible
It depends on the law of the country. Some countries (notably the United States) allows people to renounce citizenship even if it will make them stateless, although they will try to talk you out of it:
7 FAM 1215(e):
In making all these points clear to potentially stateless renunciants, the Department of State will, nevertheless, afford them their right to expatriate. We will accept and approve renunciations of persons who do not already possess another nationality. It should be noted, however, that if a foreign state deports such individuals, he or she may find themselves deported to the United States, the country of their former nationality.
Many other countries, including all the countries that are party to the Convention on the Reduction of Statelessness, do not allow renunciation unless the person already has another nationality or is going to get another nationality.
what are the consequences?
The main consequence is that it will be much harder to travel internationally without a passport, so you would be stuck in the country where you were when you made the renunciation. Some countries have non-citizen travel documents for their residents who cannot get a passport, but even if you can get that, not all countries accept those and you would probably have a hard time getting a visa even to countries that accept them.
If you were a permanent resident of the country you were in, you would be able to continue to stay and work there and daily life wouldn't change much.
If you had only a temporary status in that country, then the question is what happens when that status ends. The country will try to deport you if you don't leave on time, but where would they deport you to? The above quote from the FAM seems to say that it may be possible for someone who has renounced US citizenship to be deported back to the US; I am not sure whether the US will always accept the person back in this case (and I am not sure what status the person will have in the US if they accept them back). Your former country of nationality might have some obligation to take you back since you used their passport to enter the country (but I am not sure).