Deep in the $14.1 billion supplemental appropriations bill that President Reagan vetoed last weekend lies proof that, while Army Corps of Engineers projects may seem moribund, they almost never die.

The bill directs the corps to do a new study on the economic feasibility of a cross-Florida barge canal, a project that President Nixon scrapped in 1971 as a financial and environmental waste.

Another fine-print item is a House order that the administration must spend all $4 million that Congress appropriated for fiscal 1982 on an eastern Kentucky flood-control project, the Yatesville dam, that President Carter objected to and nearly killed.

But the big-ticket item that got Reagan's dander up was a flood-control project along the West Virginia-Kentucky border in central Appalachia, which he said would cost more than $1 billion before it is completed.

Although the final-cost figures are in question, Reagan apparently bridled at being directed by the House and Senate Appropriations committees to make certain that all the money appropriated for fiscal 1982 -- about $7.5 million -- gets spent on the Tug Fork, W.Va., project.

The water provisions of the catchall spending bill, which the president characterized as a "wasteful and unnecessary" example of congressional budget busting, are just a drop in the economic and political bucket.

Unless Congress bows to Reagan's objections or overrides his veto upon return from recess next week, funding to get dozens of federal agencies through the last month of fiscal 1982 will stop.

But the age-old grappling between legislative and executive branches over the public-works dollar, a battle that rose to fever pitch when Jimmy Carter tried to execute summarily more than three dozen congressionally beloved water projects, was spotlighted again by Reagan's veto last Saturday.

The president didn't bother to denounce the congressional proclivity for keeping favored water projects in the appropriations pipeline, but the National Wildlife Federation's water-resources director, Edward Osann, did it for him.

"These supplemental appropriations measures lend themselves to mischief of this nature. They provide an ideal opportunity for legislative insiders to push through seemingly obscure provisions that result in a perversion of public policy," Osann said yesterday.

"Tug Fork is the wrong solution to a serious problem. . . . Yatesville has been a project in search of a purpose for over a decade. . . . The Florida barge canal is truly one of the all time public-works dinosaurs," he continued.

"These projects do not reflect pressure of a strong business lobby or some big-time political action committee money. They are pets of individual members of Congress."

Tug Fork fits Osann's analysis almost to a tee. It was conceived after torrential rains blitzed the mountainous Appalachian sections of West Virginia and Kentucky in the spring of 1977.

Area residents demanded relief and protection, staging a mass march on Washington a year later, and Congress rose to the occasion.

West Virginia Sens. Robert C. Byrd, who was then the majority leader, and Jennings Randolph, who then chaired the Environment and Public Works Committee, combined with other legislators in the region to pass a bill authorizing the corps to do whatever was necessary at whatever cost to solve the flood problem.

Current estimates are that the Tug Fork project, a package that includes work on the Tug and Levisa forks of the Big Sandy River as well as local flood-protection works on the Upper Cumberland in Kentucky, will be some $700 million.

Although the supplemental contained no new money for Tug Fork, it made clear that Congress intended the administration to put aside its reservations and spend already appropriated funds on floodwalls and improvements at Pineville and Barbourville, Ky. and Williamson, W.Va.

Office of Management and Budget spokesman Edwin L. Dale Jr. acknowledged this week that Tug Fork is not authorized at the $1 billion level Reagan cited.

He said the figure "wasn't based on hyperbole" but rather on the fact that water-project cost estimates traditionally increase.