What are the largest anarchist organisations internationally by membership? I am curious about the relative popularity of different anarchist ideologies, to understand if say, anarcho-capitalism is more or less popular worldwide than anarcho-syndicalism, or something else.
This heavily depends on how you qualify "movement" and how you measure popularity.
The most active revolutionary Anarchists worldwide are:
Anarchism was also saw increased popularity in Greece after EU crisis, however there are no clear organizations with mass support that I can point to.
The societies oppose nationalism - identification with one's own nation and support for its interests, especially to the exclusion or detriment of the interests of other nations according to Oxford Languages - and support some form of direct democracy. The Kurdish movement also practices Democratic confederalism which is partially based on the ideas of American anarchist Murray Bookchin and the zapatistas were partially inspired by the ideas of social Anarchism (& AnarcoSyndicalism according to the Anarcho-Syndicalist Review article written by Anarchist Andrew Flood), so they are at least anarchist adjacent and based around libertarian socialism.
This heavily depends, but I believe one of the largest movements is anarcho-mutualism because there is a community actively practicing it called FEJUVE or The Federation of Neighborhood Councils-El Alto. As Emily Achtenberg said in her 2004 book Community Organizing and Rebellion: Neighborhood Councils in El Alto, Bolivia, the community's informal economy is most like anarcho-mutualism: an anarchist school of thought that advocates a socialist society based on free markets and usufructs - the right to have property if it is actively being occupied or used to provide goods/services for the community. Back in 2008, FEJUVE was estimated to have a population of 114,000 and the community has been around since 1979, so it has lasted for forty-three years.
In addition to the other answers, I believe the most popular kind of anarchism is democratic confederalism: a political concept of having a society based on democratic self-organization with a confederation based on autonomy, direct democracy, political ecology, feminism, multiculturalism, self-defense, self-governance and elements of a cooperative economy. It is currently being practiced in the Autonomous Administration of North and East Syria or Rojava. It has been around since 2012 and since 2018, it has a population of over 2 million. It is libertarian socialist to some, but democratic confederalism is literally inspired partially by the ideas of anarchist Murray Bookchin according to one of the founders of the ideology Abdullah Öcalan, so it is not completely disconnected from anarchism and is even described as being deeply inspired by anarchism in the text Stateless Democracy written in 2015 Renée In der Maur & Jonas Staal.