The Los Angeles Police Department was one of the first to adopt, in 1979, a "don't ask, don't tell" policy with regard to illegal immigrants. That policy, known as Special Order 40, is being challenged in court as a violation of California state law. A separate lawsuit brought by Judicial Watch is challenging the policy as a violation of Federal law.

Here is a link to the policy, as issued by the LAPD's board of commissioners. The policy forbids arresting anyone under the illegal entry provisions of the U. S. Immigration Code, and it forbids "police action with the objective of discovering the alien status of a person." It does require reporting to Federal immigration authorities when an undocumented alien is arrested on a felony, a "high grade" misdemeanor, multiple misdemeanors, or a repeat offense. There's nothing in the policy, however, that would allow the police to hold an arrestee on anything more than the non-immigration-related offense.

Federal law passed in 1996 makes LA's policy illegal:

…a Federal, State, or local government entity or official may not prohibit, or in any way restrict, any government entity or official from sending to, or receiving from, the Immigration and Naturalization Service (now Immigration and Customs Enforcement) information regarding the citizenship or immigration status, lawful or unlawful, of any individual.

The LA Times story mentions that many rank and file officers want to see the policy repealed, but they are afraid to speak out. The mayor and police chief both support Special Order 40.

The argument made in support of such a policy -- Tulsa has something very similar in place -- is that police depend on the cooperation of crime victims to do their work of protecting the public. If the police could report immigration status, some crime victims might not come forward for fear of being deported. Michael Williams offers a rebuttal:

[Illegal immigrants are] "living in the shadows" because they chose to break the law and come here illegally. All sorts of criminals "live in the shadows" because of their crimes. Drug dealers and pimps hesitate before calling the cops, too, but should we stop prosecuting them? Criminals shouldn't feel comfortable approaching the police.

When you put yourself beyond the reach of the law, you put yourself beyond the protection of the law.

UPDATE: In the comments, Roy asked for details about Tulsa's policy. MeeCiteeWurkor has a scan of the Tulsa Police Department's "sanctuary city" policy (PDF), which is even more friendly to illegal aliens than LA's Special Order 40. For example, if a citizen reports a likely illegal alien to the Tulsa Police Department, the citizen is simply to be given the number for the Bureau of Citizenship and Immigration Services enforcement office in Oklahoma City. Unlike LA's policy, there is no provision for TPD to report repeat or major offenders to the Feds. It appears that the policy was originally approved under Mayor Susan Savage in 1995, then updated in 2003 to reflect the renaming of the INS.

MeeCiteeWurkor has an extensive archive of items on illegal immigration and enforcement (or lack thereof) in Tulsa and Oklahoma.