The Supreme Court opens one of its busiest terms today, confronting an unusually large agenda of intensely political and emotional issues, ranging from abortion and tax exemptions for discriminatory schools to regulation of nuclear power and congressional authority to veto actions of the federal bureaucracy.

These are issues also preoccupying the public and other branches of government, providing striking evidence that, in the words of American Enterprise Institute court analyst Bruce Fein, the Supreme Court "has blossomed into a full partner" in the making of public policy.

As the new term begins, observers are concerned about the court's capacity to handle this role, not because of the rising caseload that has prompted complaints from several justices, but because of deepening divisions and hardening ideological positions among the justices. The result has been fragmented opinions, with each justice writing separately, and no coherent majority decision on many cases.

"Separate opinions are becoming the norm, instead of the exception," Justice John Paul Stevens, himself one of the most frequent offenders, complained to a meeting of lawyers and judges this summer.

Last term, the justices feuded in public, in unfriendly footnotes and in personalized asides and private grumbling that did not stay private very long. Much of it revolved around the newest justice, Sandra Day O'Connor, and began the day she was sworn in.

"A coronation," huffed one justice about the fuss about the first woman to join the high court. As she aligned herself solidly with the court's conservative bloc, liberals began accusing her openly of stretching aspects of a case to arrive at a predetermined result and of betraying precedents only weeks old.

By the end of the term, she was returning the fire. After a dissent in a sex discrimination case by Justices Harry A. Blackmun, William J. Brennan Jr. and Thurgood Marshall in which her womens' rights credentials were attacked, O'Connor asked them to "get off her back," as one of her acquaintances interpreted it, by writing in a footnote responding to the dissent, "We decline the opportunity to address further this ad hominem personalized argument."

The friend and other court watchers believe that some of the justices were offended by O'Connor's outspokenness, unusual for a first-year justice.

In fact, the dispute runs much deeper and illustrates the ideological crosscurrents at the court and in the law generally as President Reagan begins to make his mark on the courts. O'Connor clearly identified herself with the judiciary's own "new federalism" movement to get the federal courts out of the affairs of states and other branches of government.

Brennan, now in his 26th year, and Marshall, in his 15th, made their names helping to build the powerful federal judiciary that this movement now seeks to wither. Blackmun, though a latecomer to liberalism, wrote the opinion that epitomizes judicial activism to the conservatives: the 1973 ruling legalizing abortion.

"I've seen some bitter fighting before," said Eugene Gressman, a Supreme Court scholar and former clerk. "But it didn't surface this way in opinions. It reflects the increasingly difficult problems that face the court, the strongly held personal views from which no one wants to retreat."

Dennis Hutchinson, another former clerk now at the University of Chicago Law School, said he believes that the feuding is the worst in opinions in this century.

"They are nine strong-willed individuals who don't seem to have the time or inclination to talk to each other," he said. "They have enough trouble getting together as it is. If they are being shrill and mounting campaign speeches, they look less and less like a court and more like just another policy-making branch of government."

The court has agreed to full review of 126 cases, a record number this early in the term. Another 1,011 were considered for review during the court's secret conference Friday, with the results to be announced when the court opens.

Among those is an appellate ruling upholding constitutionality of Minnesota's tuition tax credit plan. The court's action in that case will affect the current congressional debate on the administration's tuition tax credit proposal.

The court also faces five major abortion cases, testing the right of state legislatures and city councils to place obstacles between a woman and the right to an abortion.

Two of the cases concern an ordinance enacted in Akron, Ohio, requiring, among other things, that women wait for a 24-hour "cooling-off period" before becoming eligible for abortion at any stage of pregnancy and that doctors read them an anti-abortion statement declaring that the fetus is a human life and describing its anatomical features in detail.

The Akron law, unenforced pending the Supreme Court review, also requires girls 15 and younger to obtain parental consent before having an abortion.

Two other cases from Missouri and a third from Virginia test state laws mandating hospitalization for women obtaining abortions after the first three months of pregnancy.

The Missouri statute also imposes a 48-hour waiting period, requires special pathology reports following all abortions, mandates the presence of two physicians for abortions involving a "viable unborn child" and requires parental or court consent for an abortion on a woman under the age of 18.

If upheld, the abortion laws are expected to serve as models for anti-abortion forces throughout the country. The review is the court's most thorough one since the 1973 ruling legalizing abortion.

It is also serving as a vehicle for the administration's overall attack on "judicial activism." Solicitor General Rex Lee filed a "friend of the court" brief in the cases asserting that it is time for the courts to return to the state legislatures power to regulate abortions.

The justices have also agreed to consider:

* Whether a federal law banning mailing of unsolicited ads for contraceptive devices violates the first amendment, as a U.S. District Court judge in Washington ruled in 1981 in Bolger vs. Young Drug Products Corp.

* Whether home videotaping of movies and other entertainment infringes copyrights of artists and studios that produced them (Sony Corp. of America vs. Universal City Studios). The 9th U.S. Circuit Court of Appeals ruled that it did, prompting one of the year's hardest fought congressional lobbying battles, with the videotape industry trying to nullify the ruling and the entertainment industry trying to profit from it.

* The question, left over from last term in Immigration and Naturalization Service vs. Chadha, about constitutionality of the "legislative veto," a power incorporated into more than 200 federal laws and most recently wielded when Congress voided a Federal Trade Commission regulation requiring used car dealers to disclose defects in their cars.

More expansive veto power is under consideration in the regulatory reform bill now before Congress.

* Whether the Internal Revenue Service, without express congressional authorization, may withhold tax-exempt status from private religious schools that discriminate against minorities. The IRS has been wielding this lever against discrimination since the Nixon administration and until last year, defended it against the challenges now before the court from Goldsboro, N.C., Christian Schools and Bob Jones University.

The government switched signals last year, drawing an intense political backlash from civil rights leaders and minorities. The court then appointed William T. Coleman Jr., a Washington attorney and former secretary of transportation, to argue in support of the exemption power.

* A California case, Pacific Gas and Electric Co. vs. State Energy Resources Conservation & Development Commission, testing the right of states to ban nuclear power plant construction until there is a federally approved plan for nuclear waste disposal.

* A case relevant to the Prince George's County jail rape problem about whether jailers can be sued personally for money damages for knowingly placing prisoners in cells where they might be raped (Smith vs. Wade).

* A case, brought by a man confined for four years in St. Elizabeths Hospital here after being acquitted by reason of insanity of stealing a coat, challenging D.C. procedures for prolonged commitment there. (Jones vs. U.S.).

* Whether federal laws require commercial and public television stations to make programming available to the deaf and hearing-impaired (Community Television of Southern California vs. Gottfried, and FCC vs. Gottfried).