Most of the examples which come to mind when I even try to imagine what "applying the sharia" would mean in the U.S. context are either things which are already clearly prohibited by the First Amendment or things which are clearly protected by the First Amendment. So the Oklahoma referendum fundamentally seems either redundant or unconstitutional.
The U.S. Constitution already protects non-Muslim from having "the sharia" imposed on them, while Muslims should be free to follow the sharia in matters such as marriage, inheritance, business contracts and financial arrangements. And if the terms of such agreements were drawn up in a sufficiently clear manner, why couldn't or shouldn't they be adjudicated by a U.S. court? In fact, Jews and Christians in the US already have established several institutions which allow for alternative conflict resolution according to their own religious principles but with a certain amoutn of legal validity as well. Why couldn't Muslims set up similar "courts" in the U.S.? I've never been to law school but I find it hard to imagine that a referendum which actually singles out a specific religion for special exclusion could pass constitutional muster.
Here is what appeared on the ballot in Oklahoma:
STATE QUESTION NO. 755 LEGISLATIVE REFERENDUM NO. 355
This measure amends the State Constitution. It changes a section that deals with the courts of this state. It would amend Article 7, Section 1. It makes courts rely on federal and state law when deciding cases. It forbids courts from considering or using international law. It forbids courts from considering or using Sharia Law.
International law is also known as the law of nations. It deals with the conduct of international organizations and independent nations, such as countries, states and tribes. It deals with their relationship with each other. It also deals with some of their relationships with persons. The law of nations is formed by the general assent of civilized nations. Sources of international law also include international agreements, as well as treaties.
So in addition to the sharia, the Oklahoma courts apparently can not consider the Sharia Law is Islamic law. It is based on two principal sources, the Koran and the teaching of Mohammed.
SHALL THE PROPOSAL BE APPROVED?
FOR THE PROPOSAL — YES
AGAINST THE PROPOSAL — NO
The only thing clear about the law is that it loudly says: "We hate Muslims and want to give them a hard time."
But even apart from its impact on Muslims, it seems like the proposal opens up a whole can of worms. It doesn't just try to exclude the use of the shaira but international law as well. Maybe they can start building Black site prisons in Oklahoma to replace Camp X-Ray in Guantanamo since the Geneva Convention apparently doesn't apply there? Or maybe Oklahoma will become a resort for international criminals if extradition treaties are no longer valid there? How does the referendum effect Indian casinos if Oklahoma courts refuse to respect tribal law?
CAIR has already made some legal moves against the referendum (An initial hearing is set for today.) InshaAllah sooner heads will prevail.
Ballotpedia: Oklahoma "Sharia Law Amendment", State Question 755 (2010)
Huff Post: Caliphate on the Range? The Shariah Precedent in American Courts
The American Muslim: Islamic Sharia and Jewish Halakha Arbitration Courts - updated 5/21/10