Nikah halala mandates that if a divorced Muslim woman wants to remarry her ex-husband, she must first marry another man, consummate the marriage and divorce him.
Nikah mutah and nikah misyar are temporary marriages contracted for periods between three days and three months, after which the woman is paid an agreed sum.
A five-judge constitution bench had last year ruled by a 3:2 majority that the instant triple talaq was unconstitutional.
But it had declined to rule on polygamy, nikah halala and contractual marriages, referred to it by a three-judge bench in 2015, saying these should be considered at a later stage.
On Monday, the bench of Chief Justice Dipak Misra and Justices A.M. Khanwilkar and D.Y. Chandrachud cited this history and referred the remaining issues for consideration by a five-judge bench.
Ashwini Upadhyay, an advocate, has filed the main petition while three others - Sameera Begum, Nafeesa Begum and Mohsin Bin Hussain Bin Abdad Al Kathiri - have moved intervention applications.
Upadhyay has argued that the practices under the scanner violate Articles 14 (equality), 15 (non-discrimination) and 21 (life and liberty). He has contended that nikah halala amounts to rape under Section 375 of the penal code, and polygamy is an offence under Section 494, which deals with bigamy.
He has pleaded that since these practices are illegal for Christians, Parsis, Hindus, Buddhists, Sikhs and Jains, they should be illegal for Muslims too.
Al Kathiri, a Hyderabad resident, has pleaded that "the Constitution neither grants any absolute protection to any personal law of any community that is unjust, nor exempts personal laws from the jurisdiction of the legislature or judiciary".
Nafeesa Begum has rued the absence of a law for a Muslim woman to file a case of bigamy or rape against men engaging in the practices she has challenged.
Sameera Begum says she represents the women victims of polygamy and the other practices mentioned, which she claims are contrary to the Constitution "and injurious to public order, morality and health".
"These can be superseded by the state just as it prohibited human sacrifice or (the) practice of sati.... Polygamy and nikah halala (can be declared) as an offence under (penal code) sections 498A (harassment by husband or in-laws), 375 (rape) and 494 (bigamy)," she has pleaded.