The Supreme Court said yesterday it will go ahead with its review of the politically explosive issue of whether segregated private schools are entitled to tax exemptions.

But it took the highly unusual step of appointing a special counsel, former U.S. transportation secretary William T. Coleman Jr. to argue against the exemptions. Coleman will argue in place of the government, which started one of the most intense civil rights controversies in years in January when it reversed its long-held position that the Internal Revenue Service had the legal authority to deny the exemptions as a weapon against discrimination.

Coleman, a Cabinet member under President Ford, is a veteran of both civil rights litigation and high-level corporate law. He is a senior partner in the Washington office of a major Los Angeles law firm--O'Melveny & Myers--and in 1948 was the first black law clerk at the high court.

The court will probably hear arguments in the fall and issue an opinion next year in the cases, involving Bob Jones University in Greenville, S.C., and Goldsboro Christian Schools in Goldsboro, N.C.

Thus, a policy change the administration hoped to bury with a late Friday afternoon announcement on Jan. 8 will still be a prominent source of contention at election time in November.

The schools were losers in yesterday's action. They had asked the justices to strike down the lower court ruling upholding the authority of the IRS without hearing further arguments. The court rejected that request yesterday without comment and without dissent.

William Bentley Ball, attorney for Bob Jones, said that in his view there were still only "two parties to this case: the schools and the government." With the Reagan administration no longer supporting the lower court ruling, Ball said he believed the schools should have been declared victors in the dispute.

Coleman will argue as a "friend of the court," without a specific client to represent. His mission, as described in one of the orders issued in the case yesterday, is "to argue in support of the judgments" of the 4th U.S. Circuit Court of Appeals.

"I think this is a very great opportunity and an interesting issue," Coleman said yesterday. "It'll make my summer a little more concentrated."

Appointment of a third party to argue against the schools was suggested both by civil rights organizations and the administration itself, as part of its effort to calm the legal and political waters stirred by the January policy change.

The government had successfully fought Bob Jones and Goldsboro and more than a hundred other schools denied exemptions for more than a decade. After the 4th U.S. Circuit Court of Appeals ruled that the IRS had acted properly in the Jones and Goldsboro cases, the schools appealed to the Supreme Court. The government fought them there, too, until January, when it stunned civil rights advocates with its switch.

The government's position now is that the IRS does not have the legal authority to deny the exemptions without legislation, which the administration has proposed.

It is continuing to deny exemptions pending resolution of the controversy, however. Thus, the dispute between the schools and the government, and the case itself, are still alive, but the government is unable to argue respectably against the schools any longer since it shares many of their views.

No lawyer interviewed yesterday could recall another instance in which the court had appointed an outside lawyer to represent the U.S. government in a major case. Coleman will be working on his own but will be supported by "friend of the court" briefs submitted by major civil rights organizations including the NAACP, the Lawyers Committee for Civil Rights Under Law, the American Civil Liberties Union and the North Carolina Association of Black Lawyers.

The case involved so many motions to intervene and requests for "friend of the court" status that the justices issued 11 orders yesterday to establish the ground rules in the proceeding. More are expected, because the justices must still decide whether the U.S. will be allowed to argue at all in the case.

Coleman's name had surfaced privately among civil rights lawyers in the past few weeks as the man for the job. He said he was asked Friday (by a court official he would not identify) whether he would be "available" for an assignment. Coleman said the specific case was not named in the phone call but that he was pretty sure what it was.

When asked why he thought he was chosen, the 61-year-old Republican said, "Read my biography." Coleman graduated from the University of Pennsylvania and Harvard Law School with honors, clerked for Justice Felix Frankfurter, and worked on the team of civil rights lawyers that won Brown Vs. Board of Education, the 1954 school desegregation landmark case.

As a high-priced private lawyer in New York and Philadelphia, he represented major corporations and became wealthy while continuing to serve in leadership positions in civil rights organizations. He reportedly turned down sub-Cabinet appointments, appellate judgeships and numerous other offers of government employment before agreeing to serve in Ford's Cabinet.