In the face of protests across the country, a Supreme Court challenge and the opposition of President-elect Ronald Reagan, the Selective Service yesterday began registering teen-agers born in 1962 for a possible future military draft.

Demonstrators invaded the federal building in Boston and marched in the streets of New York to protest resumption of registration. Police arrested more than 50 protesters in the two cities.

Antidraft protesters vandalized three Los Angeles area post offices by jamming toothpicks and pieces of wood into door locks before the appearance of draft registrants. As many as 100 demonstrators carrying placards picketed the federal office building in Louisville, Ky., for an hour in eight-degree cold.

Smaller demonstrations, some drawing only a handful of protesters, were staged in Phoenix; New Orleans; San Francisco; Spokane, Wash.; Seattle; Providence; Houston, Austin, Tex.; Minneapolis, and Burlington, Vt.

In several cities, young men were urged by draft opponents to register as conscientious objectors, or under protest.

Under a draft registration plan initiated by President Carter in December 1979, after the Soviet intervention in Afghanistan, and approved by Congress, young men who were born in 1962 and who will turn 19 this year are required to register for a military draft through Saturday.

In the first such registration, last July, 3.7 million of the 3.9 million eligible young men born in 1960 and 1961 have signed up for a draft. If there is a similar turnout this week, nearly all of the 1.9 million eligible young men born in 1962 will register.

After this week, males will be required to register within 30 days after they turn 18.

Even though the military draft expired in 1973, after U.S. disengagement from Vietnam, Carter proposed registration as a means of increasing military readiness. The Selective Service says registration would reduce by four weeks the time required to draft 100,000 men in an emergency.

But during his election campaign, Reagan opposed peacetime draft registration as a "meaningless gesture" that would save little time and would limit registrants' personal freedom.