The stock and bond markets staged big rallies yesterday amid signs that the Federal Reserve is not trying to stop the continued decline in short-term interest rates.

Oil prices fell sharply on the spot market, after rising significantly last week, and that added fuel to the general market rally.

Financial analysts were divided over whether they expect the Federal Reserve soon to cut its discount rate -- the interest it charges financial institutions that borrow reserves from the Fed -- but a number said that the 9 percent prime lending rate at commercial banks is apt to fall within a few days, even if the discount rate does not.

Yesterday, the relatively small Southwest Bank of St. Louis cut its prime rate to 8 3/4 percent, but no major bank followed.

The key federal funds rate -- the interest rate charged when banks lend reserves to each other -- dropped below 7 percent. From mid-January until early March, the federal funds rate averaged close to 7.9 percent.

Changes in this rate often are scrutinized by analysts for clues to Federal Reserve intentions. Some analysts believe the Fed is letting the rate decline because most evidence now available indicates the economy is still growing at a fairly weak pace.

In yesterday's market rallies, the Dow Jones industrial average climbed 34.25 points to close at 1,769.76. It was the third-largest gain of the year, and the 10th-biggest on record. The advance was also the first since last week's record loss of 82.5 points. Meanwhile, prices of government and corporate bonds also generally rose strongly.

"Any setbacks, such as we had last week, are viewed as potential buying opportunities," said economist Allen Sinai of Shearson Lehman Bros. "The fundamentals are so good for the financial markets that the drift downward in short-term interest rates, unimpeded by the Fed, helped the market shake itself out of the setback."

Sinai does not expect the Fed to cut the discount rate soon, unless similar rates again are lowered in Japan and, perhaps, West Germany. A unilateral cut in the discount rate could weaken the dollar on foreign exchange markets in a risky way, Sinai said.

The same sort of concern caused the Federal Reserve Board to postpone a discount-rate cut it had decided to make in late February for nearly two weeks while a coordinated cut was worked out among the three nations' central bankers.

The stock market "is responding again to the anticipation of declining interest rates and expectations that the Federal Reserve will cut the discount rate," said Michael Metz of Oppenheimer and Co.

"The question is, though, how much longer can interest rates alone fuel the rally?" he said. "There's already been a huge response to declining interest rates. We need something more material from the economy to fuel this rally along."

One such possible stimulus, stronger corporate earnings, continues to elude the market, he said, adding that the results tor the first quarter, now begining to come out, "will not be terribly good."

The collapse of last week's rally in crude oil and the belief that the Federal Reserve is about to lower the discount rate sent contracts for future purchases of Treasury bonds soaring, analysts said.

"The strength of Treasury bill futures is telegraphing dealers' belief that there's an imminent cut in the discount rate coming," said Gary Dorsch, senior money market analyst with G. H. Miller & Co. in Chicago.

Treasury bill futures have been on a steady rise over the last few weeks, even though bond futures have suffered some sharp corrections to their recent highs.

"So there's still a lot of optimism that interest rates are going lower," Dorsch said.

Energy prices took a nosedive on the New York Mercantile Exchange, ending a five-day rally that had lifted crude oil futures nearly $4 a barrel.

Heating oil plunged the expanded 4-cent-a-gallon limit for daily trading before trimming a bit of the loss. Crude oil for delivery in September touched the $1.50-a-barrel limit decline.

The main reason, said analyst Peter Beutel of Rudolf Wolff Energy in New York, was that the Soviet Union was reported to have signed for the first time net-back sales agreements for crude oil. A net-back agreement ties the price of the crude to the ultimate price the refiner gets for the product.

Meanwhile, in Chicago, where he was receiving an award from the University of Chicago, Beryl Sprinkel, chairman of the Council of Economic Advisers, said he believes the country's inflation rate could be down to 1 or 2 percent annually by the time President Reagan finishes his term.