The Western Wall is administered as an Orthodox synagogue. That single administrative fact generates most of the modern argument, and the argument is very largely among Jews.
The plaza is divided by a barrier into a men's section and a smaller women's section. Services follow Orthodox practice. The Western Wall Heritage Foundation and the rabbi of the Western Wall, an Orthodox appointee, set the rules.
Women of the Wall
Formed in 1988, Women of the Wall have sought to hold women's prayer services at the wall with Torah reading, tallit and tefillin, practices that Orthodox authority reserves to men and that Conservative and Reform Judaism do not.
The litigation has run for three decades and has gone both ways. Israel's Supreme Court permitted the practice in 2002 and reversed itself in 2003, upholding the prohibition on public-order grounds and directing the group to Robinson's Arch, an archaeological area south of the main plaza, as an alternative site. Members have been detained at the wall for wearing prayer shawls.
The compromise that was agreed and then shelved
In January 2016 the Israeli cabinet approved a plan for a permanent, properly built egalitarian prayer section at Robinson's Arch, with joint administration including non-Orthodox representation. For the Reform and Conservative movements, which together account for the majority of affiliated Jews in the United States and almost none of the religious establishment in Israel, the point was never the square metres. It was the recognition, at the most symbolically loaded site in Judaism, that theirs is a legitimate way of being Jewish.
In June 2017 the government suspended the decision under pressure from Haredi coalition partners. It has not been implemented. The reaction from American Jewish organisations was severe and the episode is generally treated as a turning point in the relationship between Israel and the diaspora.
What is underneath
The wall is a Herodian retaining structure, not a wall of the Temple, and the prayer plaza exposes only a short section of it. The Western Wall Tunnel runs along much more of its length underground, and excavation there has been contentious: the tunnels pass beneath the Muslim Quarter, and the opening of a northern exit in 1996 set off riots in which around eighty people were killed.
The plaza itself was made by demolition. The Mughrabi Quarter, a Muslim neighbourhood endowed as a waqf in the twelfth century, stood directly in front of the wall until Israeli forces cleared it within days of the capture of the Old City in June 1967. Around 650 residents were displaced; the number of households is given variously in the literature and is not settled. Before the demolition the prayer space in front of the wall was an alley a few metres wide.