On the bulletin board in the press room at the Supreme Court there is a one-line Associated Press dispatch. It says: "This place is an absolute goldfish bowl. There is no secrecy here."

It is a quotation from Supreme Court spokesman Barrett McGurn. It draws snickers from all who see it, because few institutions of the U.S. government make as great a fetish of secrecy as the Supreme Court.

The subject arises now because of the recent hospitalization of Justice William H. Rehnquist, one of the nine most powerful people in what some consider the most powerful branch of government. Rehnquist's hospitalization never was announced by the Supreme Court or by Rehnquist.

The same thing happened a few years ago when Justice Lewis F. Powell Jr. was hospitalized. The Associated Press correspondent at the court found out about it, not through an announcement but by discovering that Powell had failed to appear in Richmond to receive an award in his honor.

Those are examples of the court's jealous guarding of personal privacy, but the interest in secrecy goes far beyond that. Consider two examples in the last two weeks: On Tuesday, a major case on the constitutionality of restrictions on independent expenditures in presidential campaigns went unresolved by the court because of a 4-to-4 vote. Justice Sandra Day O'Connor, whose vote could have resolved the issue, had disqualified herself. As is typical at the court, she refused to explain why.

A week earlier, the court issued an opinion upholding a 40-year sentence for a man convicted of possession and sale of nine ounces of marijuana. No one except the justices knew who wrote that opinion. Had it not been for some written dissents, it would be impossible to figure out who voted for it. Beyond that, the court never heard full arguments on the case. This practice--the "per curiam" opinion--is roughly analogous to Congress' refusing to disclose either the sponsor or the voting on a major piece of legislation, and it is more and more frequent in this "goldfish bowl."

The phenomenon has been called "judicial lockjaw." Few dispute the need of the justices to conduct their deliberations in a secret conference. Few quarrel with their unwillingess to elaborate on their opinions off the bench. But "judicial lockjaw" goes much farther than this.

The policy most often is explained by saying that openness is unbecoming for justices and the court, or that it is unethical or inappropriate or that it prejudices cases. Or, as scholars Arthur S. Miller and D. S. Sastri--critics of secrecy--have expressed it, the court requires a "semi-holy" treatment; there is "something untouchable" about the institution that affects all American lives sooner or later, in one way or another, often with profound impact.

In 1979, Chief Justice Warren E. Burger tried to explain the reticence. Reacting to a particularly raucous encounter with television news reporters, Burger said through McGurn that "there is a long tradition of isolation of judges from day-to-day controversy. That tradition means not only that judges do not take part in political affairs or other public controversies but that it is inappropriate for them to be involved with matters not relating to the administration of justice."

There are questions about whether Burger or some of the others really believe in that explanation or whether because they have life terms and are unaccountable to voters, they are doing what elected officials wish they could do: talk when they feel like talking and shut up when they feel like shutting up.

Burger says justices should not engage in off-the-bench controversy. But last winter he stirred up a storm by delivering a speech in favor of hard-line law-and-order policies. On Monday, the speech came back to haunt him when a lawyer arguing a case before the court cited it in support of his position as if it were case law. Rehnquist gave a speech the previous year denouncing the tendency of judges to intervene in family affairs. In 1979, Justice William J. Brennan Jr. offered the public an off-the-bench interpretation of the First Amendment rights of the press.

Here are some examples of things they generally choose not to talk about:

When they issue orders, often affecting millions of dollars or individual liberty, they often decline to explain why.

With some exceptions, they choose not to explain why they disqualify themselves from individual cases. One justice offered reporters a written explanation recently on the condition that it not be disclosed to the public.

Last year, Justice Harry A. Blackmun declined to explain his unusual action withdrawing from a major case at the last minute after fully participating in oral arguments. The withdrawal, from an opinion holding that Congress unconstitutionally reduced judicial salaries on two occasions, was so late that one justice recalls "some of us sitting there with our mouths open" when they heard about it. Sources said Blackmun was protesting the timing of the opinion, but he refused to comment.

Justice Powell repeatedly declined to comment last year when the Des Moines Register alleged that he was voting on some cases where he had a conflict of interest.

Rehnquist totally declined comment on his health. Burger flatly refused to comment on allegations made against him by John D. Ehrlichman that he discussed pending cases with President Nixon, even getting into a televised fracas with a Nebraska television cameraman to avoid being asked questions about it.

Inconsistencies abound. Burger, in an opinion last year, endorsed experiments in the states to permit television coverage of courtroom proceedings even though opponents argued that the presence of cameras could influence juries and witnesses. But he has refused to allow any tape recording or televising of oral argument at the Supreme Court, where there are no juries and no witnesses.

In an opinion the year before, Burger enthusiastically endorsed the principal of press and public access to the system of justice in his opinion requiring that most trials be open. But the meetings of the policy-making arm of the federal judiciary, the Judicial Conference, which is headed by Burger, are closed tight.

The serious examples of secrecy are numerous. The petty ones are multitudinous.

A few months ago, someone distributing Supreme Court office telephone directories inadvertently placed one on the desk of each reporter. Upon discovering this, a court security official swept through the press room, rummaged through desks and shelves and seized them. Officials later apologized to reporters. But the telephone directories--which provide nothing more than the switchboard operator routinely provides--never were returned.

At the time of Sandra Day O'Connor's swearing-in, the court forbade interviews with the carpenter who had built the new chair. "Our policy," McGurn said simply at the time, "is that no one here gives interviews."

Even retired justices get nervous about talking publicly. Former justice Potter Stewart, who has held two on-the-record press conferences since leaving the bench, declined recently to comment on why as a justice he would never comment.

The mood at the court makes it one of the strangest journalistic beats in the city. Even the press sometimes behaves differently than it does on other assignments.

Last term, for example, a United Press International correspondent obtained an opinion three days before it was issued by the court. That is an extraordinary breach of court security. McGurn complained to UPI, which then killed the story, explaining that editors had doubts about the authenticity of what they had.

On most beats, whether or not such a scoop served any useful purpose, a reporter getting it would be the envy of competitors. At the Supreme Court, some members of the press appeared as outraged as a justice about the leak.

Another example: when news of Rehnquist's hospitalization broke, along with the information that his medication had been slightly impairing his speech, news organizations uniformly reported that his speech had appeared impaired on the bench for some time.

None of the news organizations, including The Washington Post, had reported the impairment before the hospitalization.