Three U.S. Army soldiers were killed and 16 others were injured today when a rocket motor for a Pershing missile caught fire and burned as it was being unpacked at an American nuclear base in West Germany, U.S. Army officials said.

It was the second mishap involving a missile in Europe in the past three weeks. On Dec. 28 an unarmed Soviet missile being used as a target went off course and flew over Norway and Finland. A Norwegian official quoted the Soviet ambassador as saying it was an "old cruise missile" that had been fired from a ship in the Barents Sea.

A U.S. Army spokesman, reflecting apparent sensitivity to public attitudes toward the Pershings, said no nuclear weapons were involved in today's incident and that no explosion took place. At no time, he added, was there any danger to the German civilian population.

The accident occurred when an Army crew, working in a tent, tried to use a special crane to lift the first of two rocket motors from a shipping container that had just arrived from the United States, U.S. officials said. The crane apparently broke, dropping the motor and igniting its contents of solid fuel.

A U.S. Army statement said the crew had followed "authorized procedures" in what was called "a routine operation" at Camp Redleg, near Heilbronn, where 36 of a total of 108 Pershing II nuclear missiles are being deployed by the U.S. Army's 56th Artillery Brigade in West Germany. Battalions in Neu Ulm and Schwaebisch Gmuend are also being equipped with 36 new missiles each.

Brig. Gen. Raymond E. Haddock, the commanding officer of the three Pershing II bases operated by the 56th Artillery Brigade in West Germany, ordered a full investigation to determine the cause of the tragedy.

"I am saddened by the loss of these fine soldiers who were performing duty in the service of their country," the general said. "I extend my very sincere sympathy to the families of the soldiers involved." The Army said it would not release the names of the soldiers until their families are notified.

Two of the soldiers were killed immediately when the first stage motor burst into flames and another man died en route to the hospital, Army officials said. Of the 16 soldiers who were treated, nine were kept in the hospital.

The first battery of nine Pershing II missiles, which can strike Soviet territory within 14 minutes, became operational in West Germany in December 1983. NATO sources said that at least 54 of the rockets are now fully deployed.

The engine fire today was the most serious accident involving the controversial new missiles since they began arriving in West Germany more than a year ago in spite of massive protests by antinuclear demonstrators.

The only other accident, according to Army officials, took place in September when one of the Pershing transport trucks, carrying an unarmed missile, tipped over into a ditch after its driver had gotten out to inspect a muddy road during maneuvers. Even though the missile carried no warhead, the accident sparked a furor when protest groups disseminated rumors that the region had become contaminated by nuclear radiation.

The swift and accurate Pershings comprise part of NATO's new arsenal of medium-range missiles designed to counter the buildup of Soviet SS20 nuclear-tipped rockets targeted on Western Europe.