All is not quiet with far-flung US territories either . . .

Even as the current battle over the Falkland Islands rivets world attention, problems facing US island territories continue largely unnoticed.

While no Falklands-type crises loom in these far-flung areas -- scattered from the US Virgin Islands in the Caribbean to the Palau Islands just east of the Philippines -- all is not quiet either.

Three issues, especially, are rumbling with increasing intensity:

* The antinuclear movement. In the South Pacific, it is gathering so much steam that former US Ambassador to Fiji William Bodde Jr. has called it ''the most potentially disruptive development for US relations with the South Pacific.''

* New federalism. President Reagan's plan would shift several federally administered programs onto the islands, whose economies might or might not be able to shoulder the burden.

* The Caribbean Basin Initiative. One of President Reagan's major pieces of legislation, it is an attempt to use direct aid and targeted tax and trade benefits to pump an additional $350 million more into the lagging economies of Caribbean nations. While Congress wrangles with the foreign aid package, the US territories in the Caribbean are concerned that it will dull their own carefully built-up economic edge.

Virtually all US territories, including the Louisiana Purchase that is now the American South and the Northwest Territory that is now the Midwest, were originally kept for defense purposes. So it is with outlying areas, says says Senate Energy and Natural Resources counsel James Beirne. The military plays a key role, which makes the antinuclear movement all the more touchy.

At one extreme is Guam jockeying for more military facilities to replace the phasing down of the Polaris submarine program based there. On the other hand, the Constitution passed by the Palau Islands (Belau) in 1980 prohibits nuclear weapons.

According to Ched Myers, the North American representative of the Pacific Concerns Resource Center in Honolulu, nuclear-free groups are working closely with independence and environmental groups on Pacific Basin issues.

''We see (Palau) as a concrete model of where Pacific islands can go,'' says Mr. Myers. There, groups for democratic self-determination fused with the nuclear-free zone movement into the coalition working against passage of the Compact of Free Association. For 13 years, the US government has been negotiating to move the Trust Territory of the Pacific Islands from US custodianship under the United Nations to Free Association status.

Although the move is supposed to allow for more self-government, the antinuclear coalition opposes it because the US would maintain exclusive military rights. When government negotiations are completed, it will be voted on by the three Micronesian legislatures.

''If the compact doesn't pass, it will provoke a crisis that will make the critical issues clear,'' Myers says. ''It's bigger than just 'get rid of the nukes.' It's a question of the political integrity of the Pacific Basin'' -- whether local governments or the United States really controls the area.

It's also a question of providing ''depth'' for the US maritime defense strategy, says Gil Slonim, a former US Navy strategic planner. The territories provide back-up bases, port facilities, supplies and repair stops for the Pacific fleets.

Because of the geopolitical isolation of the Pacific, industrial nations see it as an easy place to dump the nuclear waste that is a politically messy at home, Myers says. It becomes oceanic pollution for the people who depend the most on the ocean, he says, and the islanders see themselves left with other nations' sticky problems.

Islanders are also concerned with the potential impact of some of President Reagan's policies.

His new federalism, for example, could have a wrenching effect on the economies of the territories, which have come to lean more heavily than most states on the federal government, Mr. Beirne says. Many details would have to be worked out.

For example, a Guam delegation of businessmen, which visited the President in April, was doubtful the island needed to meet requirements of the Clean Air Act. Roger Stillwell, a staffer for Guam Congressional representative Antoia Borja Won Pat said it would cost $20 million for Guam businesses to install highly sophisticated scrubbers that would purify smokestack belch. But in Guam, the trade winds blow any smoke out to 1,500 miles of Philippine Sea.

The delegation's visit did elicit promises from the administration for new legislation easing the burden on territories.

But the territories are targeted for budget cuts along with almost everybody else. The Reagan administration says the savings will be in more efficient bureaucracy.

For instance, the administration wants to consolidate all territory food stamp, school lunch, and Aid to Families with Dependent Children payments into one territorial nutrition block grant to be administered at the local instead of the national level. The proposed grant is $869 million, compared with 1981 nutrition assistance of $968 million.

Proposed budget cutbacks spurred the banding together last fall of Washington representatives of the territories into the Territorial Caucus. ''(The federal government) will pick up the slack because we're responsible for the territories ,'' Beirne says. ''We're going back to where we used to be. Whether that will be done cleanly and without dislocation, I don't know.''