Britain and Argentina are at daggers drawn, and the outcome, at this writing, is acutely in doubt. But the Falkland Islands crisis already has told us something of lasting significance about the often-celebrated Anglo-American "special relationship."

To the considerable degree that it rests on British political trends and public sentiment, rather than the close and cozy ideological kinship between Ronald Reagan and Margaret Thatcher, its underpinnings are in a parlous state. This is all the more disquieting to American observers here when they consider the potential for the Falklands affair to make the relationship immeasurably more precarious.

The evidence is everywhere that while the Thatcher government and responsible opposition party leaders may comprehend the imperatives for "even- handedness" in the negotiating efforts of Secretary of State Alexander Haig, the populace, by and large, does not. Even those public figures now expressing an understanding of what the Haig mission was all about are unlikely to be charitable if the confrontation is resolved on terms that fall short of popular expectations.

And the public's expectations, polls show, have been inflated beyond any reasonable prospect by the government's unyielding hard line at the start.

That's the nub of it, for the Thatcher government and for U.S.-British relations. Nobody in the thick of it sees any hope for a return to the conditions before the Argentine invasion of the Falklands. Any settlement will advance Argentina's interest in ultimate sovereignty at the expense of some weakening of Britain's claim to, and hold on, the islands--and that's taking the bright side. A protracted standoff or a costly shoot- out could be far more damaging.

In any event, while the Thatcher government would bear the brunt of any result that is perceived as failure, big or small, you can be confident there will be no end of second-guessing of the American role. "Just about everybody is ready to blame us," says one American expert here.

That's what the Falklands crisis has laid bare: an all-too-easy disposition on the part of the British to distrust the United States as ally or protector. This collapse of confidence had been there all along, but somehow was unrevealed to most Americans in the rosy glow of good feelings between the Reagan administration and the Thatcher government.

More than a year ago, Market Opinion Research International (MORI), one of Britain's leading pollsters, sampled views on whether "the future of Britain should rest mainly with the Commonwealth, the United States, Europe or none of them."

The winner, with 33 percent, was "none," an outcome that Robert Worcester, head of MORI, attributes to the powerful attraction among today's Britons of what is called the "little England" view of Britain's world role.

The runner-up was Europe (27 percent), next came the Commonwealth (25 percent) and "don't know" (9 percent). Dead last was the United States with 6 percent--not exactly a solid base for the sort of "special" Anglo-American relationship that British and American leaders speak of in after-dinner toasts.

Further to the point, a Gallup poll just before the Falklands invasion found that only a narrow plurality of those Britons sampled (46 percent to 44 percent) had a favorable opinion of the United States. Britain's opinion of the United States was the lowest of five European allies, behind France (55), West Germany (73), Italy (63) and Belgium (49). The same sampling placed the British lowest among the five in their "confidence in the United States to deal wisely with world problems."

And still further, the latest MORI poll shows the Conservatives and the Labor Party narrowly ahead of the Alliance (an amalgam of the old Liberal Party and the new Social Democrats). This represents a steep drop for the Alliance from last year, with Labor moving up even with the Conservatives. Though an election is not likely anytime soon, whatever the Falklands outcome, any Labor resurgence does set one to thinking about what it connotes. Labor's platform promises to de-nuclearize Britain, leave the Common Market and generally lighten the United Kingdom's NATO involvement.

None of these trends and tendencies might matter, of course, if the Falklands crisis is brought to a reasonably acceptable conclusion, and Mrs. Thatcher can ride out a continuing economic storm. Even if her own party should turn on her, her replacement could command a considerable Conservative majority in Parliament for two more years.

But there is more than transitory anger in the attitudes toward America unleashed by the Falklands affair. The tide of British public opinion, if the analysts have it right, is not running in a way that bodes well for a smooth-working, let alone a "special," relationship of the sort the two governments' political leaders regularly invoke.