According to the website, Muslim marriages were legislated under the neatly-Orientalist category of ‘Mohamedan Marriage Ordinances’ in the 1880s, when we were under colonial rule. Edward Said, in his 1978 work 'Orientalism', called the use of Mohamedan as Orientalist because Western observers of the Orient likened Prophet Muhammad as a key figure in Islam to Jesus (or Christ) as a key figure in Christianity.
Happily, in the newly-formed Singapore government in 1957, these 'ordinances' were renamed to the 'Muslims Ordinance Act'. Uncommonly, Singapore has two parallel systems of law for Muslims and non-Muslims for the realms of marriage, divorce and inheritance. (The only other country I can think of that has equally comprehensive laws for different groups is maybe Malaysia, and maybe India.)
The AMLA (Administration of Muslim Law Act) passed in 1966 meant that Muslim marriages and divorces are conducted under the Shari'ah Court. Now, since 1978, Muslim marriages and divorces are enacted under ROMM, which is separate from ROM.
Interestingly, the Women’s Charter that was passed in 1961 by the Singapore parliament to protect rights of women and girls does not cover Muslims. That means that the legal equality of husband and wife, and the ability for battered spouses to gain protection does not cover Muslims. However, I imagine that if Muslim marriages are also registered under civil law, these laws could also apply.