New York's Lincoln Center is humming

The holiday season is booming here, and much of the action is swirling around Lincoln Center, where shows of all sorts are jostling for space. The buildings might seem crowded if one attraction - the Big Apple Circus - didn't prefer setting up quarters in its own tent.

The prima donna of Lincoln Center at Christmastime is, predictably, the New York City Ballet's annual production of ''The Nutcracker.'' This year's edition, running through Jan. 2 at the New York State Theater, finds the spectacle in full bloom, as dazzling to the eye as anything I've seen on a stage.

It's hard to avoid a cliche like ''sheer magic'' when describing the first act, with its old-fashioned story decked out in high-tech lighting and visual effects, culminating in a ''Dance of the Snowflakes'' amid an elaborate onstage blizzard. Act II is more modest, physically, though more sophisticated in its dancing. In short, it's a rousing evening, and my children enjoyed it almost as much as I did, though they disagreed on which of the two acts was more fun.

A less expensive attraction - in fact, it's free - is the 12th annual Community Holiday Festival. It continues through Jan. 4 at Lincoln Center, and features a series of children's matinees as well as family shows in the evening. The performers come from New York City's five boroughs and nearby Westchester County, presenting programs of music, theater, and dance.

This year's matinee series features the Iron Clad Agreement theater company, whose production of ''Thomas A. Edison'' depicts the inventor as he creates the lightbulb, the record player, and the movie camera. And the audience gets to participate.

The festival's family portion opens the day after Christmas with a ''Salute to the Arts and Culture That Began in Harlem,'' presented by the Harlem Cultural Council.

Back for a second year in its warmly heated tent is the Big Apple Circus, pitched as before in a comfy niche adjacent to the edifices of Lincoln Center. As the kids and I discovered at its debut last year, this isn't the biggest show on earth, with just one ring and a limited number of seats to boast of. Accordingly, the troupe puts a premium on intimacy, placing its energetic performers almost in the audience's lap.

The 1982 edition, running through Jan. 2, features new acts by some of last year's contingent, including the Back Street Flyers on a ''Russian swing,'' the Tinsmans on an ''aerial ladder,'' and the literate jugglers Paul and Michael. Also present will be such ''international circus artists'' as the Flying Gaonas of Mexico, the Dymeks of Poland, Katja Schumann and her palomino stallions from Denmark, a trampoline act, an elephant, and a Japanese performer named Koma Zuru , whose family has reputedly been performing a samurai act for more than eight centuries. Sounds exotic, to say the least.

Coming up later this season will be another centerpiece of Lincoln Center holiday fare, the annual production of Humperdinck's ''Hansel and Gretel'' by the Metropolitan Opera. Also, the Little Orchestra Society will continue its latest series of ''Happy Concerts for Young People'' on Jan. 15 with a program called ''Toy Symphonies.'' The highlight will be Rossini's ''La Boutique Fantastique,'' illustrated by artist Lisl Weil drawing larger-than-life sketches across the stage.

Sharing the bill is a salute to toys, with music by such composers as Debussy and Victor Herbert, conducted by the capable Dino Anagnost. If it matches the sprightly ''Amahl and the Night Visitors'' recently offered by the Little Orchestra group, it should be a winning program.