North Africa, South Asia and Southeast Asia want sharia courts to decide family law issues such as divorce and property disputes, the Pew Forum on Religion and Public Life said on Tuesday.
Views on punishments such as chopping off thieves' hands or decreeing death for apostates is more evenly divided in much of the Islamic world, although more than three-quarters of Muslims in South Asia say they are justified.
Those punishments led to the sharia becoming controversial in some non-Islamic countries, where some critics say radical Muslims want to impose it on Western societies, but the survey shows views in Muslim countries are far from monolithic.
"Muslims are not equally comfortable with all aspects of sharia," said the study by the Washington-based Pew Forum. "Most do not believe it should be applied to non-Muslims."
Unlike codified Western law, sharia is a loosely defined set of moral and legal guidelines based on the Koran, the sayings of prophet Mohammad (hadith) and Muslim traditions. Its rules and advice cover everything from prayers to personal hygiene.
Politics and violence
More than four-fifths of the 38,000 Muslims interviewed in 39 countries said non-Muslims in their countries could practice their faith freely and that this was good.
This view was strongest in South Asia, where 97% of Bangladeshis and 96% of Pakistanis agreed, while the lowest Middle Eastern result was 77% in Egypt.
The survey polled only Muslims and not minorities. In several Muslim countries, embattled Christian minorities say they cannot practise their faith freely and are in fact subject to discrimination and physical attacks.
The survey produced mixed results on questions relating to the relationship between politics and Islam. Democracy wins slight majorities in key Middle Eastern states - 54% in Iraq, 55% in Egypt - and falls to 29% in Pakistan. By contrast, it stands at 81% in Lebanon, 75% in Tunisia and 70% in Bangladesh.
In most countries surveyed, Muslims were more worried about Islamist extremism than any other form of religious violence. Suicide bombing was mostly rejected, although it won 40% support in the Palestinian territories, 39% in Afghanistan, 29% in Eygpt and 26% in Bangladesh.
Sex and veils
Three-quarters say abortion is morally wrong and 80% or more rejected homosexuality and sex outside of marriage. Views on whether women should decide themselves if they should wear a headscarf vary greatly, from 89% in Tunisia and 79% in Indonesia saying yes and 45% in Iraq and 30% in Afghanistan saying no.
Majorities from 74% in Lebanon to 96% in Malaysia said wives should always obey their husbands. Only a minority saw Sunni-Shi'ite tensions as a very big problem, ranging from 38% in Lebanon and 34% in Pakistan to 23% in Iraq and 14% in Turkey.
Conflict with other religions loomed larger, with 68% in Lebanon saying it was a big problem, 65% in Tunisia, 60% in Nigeria and 57% in Pakistan.