The Supreme Court awarded the bitterly contested site in the northern town of Ayodhya to Hindus on Saturday, dealing a defeat to Muslims who also claim the land that has sparked some of the country's bloodiest riots since independence.
In 1992, a Hindu mob destroyed the 16th-century Babri Mosque on the site, triggering riots in which about 2,000 people, most of them Muslims, were killed, but no major violence was reported after the court ruling on Saturday or on Sunday
About 37 people were arrested and 12 cases were registered in Uttar Pradesh, India's most populous state and the site of the contested land, state police said.
At least one person was arrested in the state capital of Lucknow for making "inappropriate remarks" on social media and using threatening language.
"Police are appealing to residents to not misuse social media," Kalanidhi Naithani, senior superintendent of police in Lucknow, said late on Saturday.
In another part of the state, at least seven men were arrested for setting off firecrackers or creating disturbances while distributing sweets in celebration, police said.
The Home Ministry did not immediately respond to a request for information on arrests.
Before Saturday's verdict, the government deployed thousands of members of paramilitary forces and police in Ayodhya and other sensitive places. Hindu groups told members not to celebrate publicly.
The court's decision paves the way for the construction of a Hindu temple on the site, a proposal long supported by Prime Minister Narendra Modi's ruling Hindu-nationalist party.
Hindus believe the site is the birthplace of Lord Ram, a physical incarnation of the Hindu god Vishnu, and say the site was holy for Hindus long before the Muslim Mughals, India's most prominent Islamic rulers, built the Babri mosque there in 1528.
The Supreme Court called the 1992 demolition of the mosque illegal but handed the plot of 2.77 acres (1.1 hectares), about the size of a soccer field, to a Hindu group. It directed that another plot of five acres in Ayodhya be provided to a Muslim group that contested the case.
Some legal scholars and Muslim activists saw the judgement as unfair, particularly given that the 1992 razing of the mosque was deemed illegal.
"Why has the 2.77 acres been gifted to the very elements who were party to this?," Syeda Hameed, the president of the Muslim Women's Forum, wrote in the Hindustan Times, referring to the mosque's destruction.
Muslim leaders have called for peace between majority Hindus and Muslims, who constitute 14% of its 1.3 billion people.
Some seemed to have resigned themselves to the decision."I am disheartened with the verdict and leaving it to Allah," said Mohammad Azam Qadri, a community leader in Ayodhya.