THOMAS ANDY BAREFOOT, convicted killer of a police officer, did not die in a Texas prison Tuesday morning as scheduled. As always happened in old Jimmy Cagney movies, the reprieve came at the last minute, only 11 hours before the death sentence was to be carried out. The U.S. Supreme Court stopped the execution, at least for a few months, in order to hear arguments and reach a decision on the procedural aspects of death penalty appeals.

Capital punishment was reinstated by the Supreme Court in 1976, and since that time over 1,000 convicted murderers have been sentenced to death. Only six have actually been executed, however, and all but two went to their deaths willingly, having abandoned final legal appeals. Last month, Charlie Brooks, convicted of urder in Texas, fought his sentence all the way to the Supreme Court. In appeals in which he was represented by 12 lawyers, Mr. Brooks had nine separate hearings and his case was reviewed by 23 judges, state and federal. The Supreme Court refused to stay his execution, and Mr. Brooks died by lethal injection on Dec. 7.

The appeals in the Barefoot case followed the same pattern--first an appeal of the conviction through the state system to the U.S. Supreme Court, then a habeas corpus petition in the federal courts. This time the high court, perhaps in deference to the three justices who dissented in the Brooks case, decided to postpone the execution until the review standards can be clarified. The specific question here is whether the Fifth Circuit Court of Appeals was required to issue a stay and to hear arguments on Mr. Barefoot's habeas corpus petition even though, on the basis of papers filed, they believed the petition to be without merit.

The court is grappling with two overriding problems in this case. The first is a refinement of a standard for imposing this ultimate, irrevocable penalty. If it is to be allowed at all, then surely defendants must have every opportunity to appeal convictions, and the courts must not act or appear to act in a peremptory manner. Still, at what point do numerous appeals become simply a device to subvert the decision of judge and jury and to make the death penalty a practical impossibility?

The justices must also consider the respective roles of the state and federal courts. Some justices clearly favor granting greater deference to the state courts in criminal cases while others view the federal tribunals as the proper court of last resort, exercising a supervisory review power over state systems. Some would restrict the availability of federal habeas corpus in the case of state prisoners; others believe that any such limits would be unconstitutional.

Judges, lawyers and legislators await the Barefoot decision because it will deal with and perhaps resolve some of these important constitutional questions. Others wait without the cool detachment characteristic of scholars. They are the thousand men and women on death row across the country.