Despite international conventions with provisions against statelessness, how does a person lose their only citizenship, or be born without citizenship, today in this day? Is it possible? What are the factors that affect a person becoming stateless?
Wikipedia has a section about it on the statelessness article. But to summarise it, the main reasons are
Conflict of laws for example, if you are born in a country which does not recognise the Jus Soli (getting nationality from the place where you are born) and the country of your parents does not recognise the Jus Sangui, inheritance of nationality.
Gender discrimination 27 countries to not recognise the right for women to transmit their nationalities.
Discrimination some countries refuse to recognise a nationality to some ethnicity. Even if that's against international treaties.
State succession if your state cease to exist. See Anixx's answer for USSR, or Yugoslavia.
Administrative in some cases, the administrative path to get the nationality one is entitled to may be complex, or costly. For example to provide a valid Birth Certificate might be a problem in some countries.
Renunciation simply having one nationality and asking to drop it.
Non-state territories though rare, there are some territories which did not belong to any states.
Probably the best answer comes from UNHCR:
Areas that have seen large movements of people due to war or economic migration have also been significantly affected by statelessness. In West Africa, the estimated stateless population in Côte d'Ivoire is 700,000, many of whom were migrants of Burkinabé descent who were not eligible for Ivorian nationality after the country's independence from France in 1960.
Statelessness due to the dissolution of former states also continues to affect many people, including some 600,000 people in Europe alone. For example, Montenegro, which was formerly part of the Yugoslav federation, has approximately 3,200 registered stateless people, while Estonia and Latvia have some 86,000 and 262,000 stateless people respectively.
In many countries stateless persons are frequently denied a range of rights such as identity documents, legal employment, education and health services. In some other settings, such as Estonia and Latvia, they enjoy rights comparable to citizens in many respects.
There have been notable examples where, through political will, it has been possible to resolve large protracted situations of statelessness. For example, the case of some 300,000 Urdu-speakers (sometimes referred to as Biharis), was resolved in Bangladesh in 2008. Similarly, the situation of the Brasileirinhos Apatridás, stateless children born to Brazilian parents abroad who were unable to acquire Brazilian nationality unless they went back to live in Brazil, was resolved in 2007. Estonia and Latvia have also both recently taken steps to further facilitate the acquisition of citizenship by those born in Estonia and Latvia to non-citizen parents, which will help ensure that these situations are resolved over time.
If the reports I read are accurate, a combination of Trump getting elected and then being born in the US if your parents are illegal immigrants would do it.
To clarify that this is indeed an answer even if an unpopular one, see https://www.bostonglobe.com/news/politics/2015/08/16/trump-would-deport-children-illegal-immigrants/0k8tnLoq6mlOygt3oGV67M/story.html
The parents country, if identified, could decline to confer citizenship.