HER HAIR is by Sassoon; her costumes by the man who dresses Shirley MacLaine, Tanya Tucker, the Lolita of country music, wants to set her up with Dino's friend, Desi Arnaz Jr. And TV heartthrob Erik Estrada has been hanging out around the rink.

Has figure skating gone Hollywood? Will she make the cover of People magazine?

"Oh, God, I'd die," said Linda Fratianne, the United States and world figure skating champion. "It's just kind of fun knowing somebody famous."

Most skating experts, including former champion and ABC commentator Dick Button, agree that Linda Fratianne has "a superb chance" of becoming the Olympic figure skating champion next winter in Lake Placid. The chances are equally good that she will become, if she is not already, more famous than say, even Erik Estrada.

Last April, when she returned from winning the world championship in Vienna, a television crew from the CBS children's series, "30 Minutes," visited Fratianne at her home in Northridge, Calif. "The lights were giving me migraines," she said. "And my coach, Frank Carroll, said, "You'd better get used to it now it's going to get a whole lot worse in a year.'"

Figure skating has become a media event. Doug Wilson, who produced ABC's coverage of the 1976 Olympic figure skating events, estimates that skating coverage has doubled in the last 10 years. When ABC began televising the world championships in 1962, and the nationals two years later, the coverage represented a small segment of a larger show. "Today," said Wilson, "skating is something people want to watch for an hour and a half."

Dick Button, who has been an ABC commentator since 1962 said, "Television has had an incredible impact on skating. There are 20 million people watching now instead of 5,000."

Fratianne, who says she "likes being on TV" and "trains harder" when she knows an event is going to be televised, is aware of the new audience:

"Twenty years ago," she said, "if I had been champion and you said the name Linda Fratianne, the public would have gone, 'Linda who?"

Today, "specials," showcasing professional skaters, like the recent "Dorothy Hamill at Lake Placid," insure that this recognition will last long after an amateur career is over.

Fratianne's amateur career will end next February after she tries to fill the skates of Dorothy Hamill, the media darling of the 1976 Winter Olympics. Hamill was, in Frank Carroll's words, "Miss Whoop-de-doo, the All-American cheerleader on ice," who adorned every photograph with a perky little smile. Fratianne (who finished eighth in 1976) idolized her, perhaps because of the easy vivaciousness behind those smiles, which, Fratianne admitted, "has always been the hardest part of skating for me."

Ironically, "Linda is what everybody thought Dorothy was," said Wilson. "She really is the girl next door with the terrific well-adjusted family. She's a wholesome, huggable, unpretentious 18-year-old kid."

Unfortunately, as one skating pro put it, "those are wonderful qualities everywhere but on the ice," where these days, "you've got to bubble up and 'strut your stuff.'"

All sports are supposed to be entertaining, or at least that's what the promoters hope. Figure skating is unique because it judges its champions not only on the basis of their technical ability but also on their ability to entertain. There are no tapes to cross or records to break, only judges to impress. They issue two sets of marks for freestyle skating, one for technique and one for what is called "artistic presentation."

"Physical presentation is beginning to overpower technique," said Sonya Dunfield, the captain of the 1952 U.S. Olympic team and now a teaching pro in New York. "Most of it has been encouraged by TV. The amateur association get lots of money for the television rights, and they want it."

Dick Button argues that, "Television did not create the enthusiasm for showy moves and showy costumes. It merely amplified elements in skating that have always been there."

Wilson believes that, "Media exposure was probably responsible" for the 1972 decision to change the formula for scoring, making freestyle skating more important than the eight school figures. The freestyle portions of the singles competition (short and long programs) now comprise 70 percent of the final score and school figures 30 percent (school figures counted for 60 percent in the past). "They finally realized that no one was going go tune in to watch someone trace school figures in the ice, said Wilson.

Freestyle skating has become decidedly more free over the years. In the 1950s, said Carroll, "If you did some of the acrobatic, emotive stuff they're doing now, you'd have been booed out of the rink. Girls never did things like butterflies."

Or triple jumps, for that matter. Fratianne, who opens her competitive program with a triple toe loop and a triple sulchow, was the first woman champion to master triple jumps and has made them virtually mandatory for all world-class competitors.

But even triples are not enough. "These days, you've got to have a gimmick," said Sonya Dunfield. Fratianne's gimmick has always been what Carroll calls "her way-out costumes," beaded, sequined, theatrical getups designed with television in mind.

"It would be great if they all went out and skated in black leotards," said Carroll. "But skating is very much influenced by glamor. That's why I'm so careful about the way Linda is groomed, why her hair is done at Sassoon and her costumes are bright red for TV.

"In the old days, not all that many people were seeing what the champion looked like," he continued "The championship was held in some dingy arena in Cleveland and then they read in the paper the next day, 'The champion skated to Swan Lake in a white dress.' "Today, they're watching on TV and they're saying, 'Her hair is different,' or, 'Ugh, I can't stand that dress.' One of the first things the judges ask me now is, 'What's she going to wear next year?' They've become much more interested in the visuals because of TV."

Fratianne, who is the object of all this scrutiny, said, "Skating is kind of like a beauty pageant. Sometimes I think, 'Why don't they start judging the skating instead of who has more Guccis and Diors.'"

It is no secret, as Fratianne said, "that appearances matter in this sport." Perhaps the most compelling statement of just how much they matter is that five of the top women figure skaters in the country, including former champion Peggy Fleming, and the current one, Linda Fratianne, have had plastic surgery on their noses in the last two years.

Of course, this may have something to do with keeping up with the Joneses. But, it may also have something to do with the fact that "they're worried about how they're going to look in closeups on TV," said Doug Wilson.

Fratianne's surgery was only partly cosmetic. Doctors also removed her tonsils and corrected a badly deviated septum in her nose. As a result, she is literally breathing easier on the ice and has more stamina for the four-minute freestyle program required in competition.

It is always awkward when something as personal as plastic surgery becomes the object of public scrutiny. Fratianne betrays little self-consciousness in discussing it. "When you come from L.A.," she said, "it's no big deal.

"I didn't do it for the judges or anybody else. I did it for myself. I think this is the way I'm supposed to look. God just made a mistake on me the first time."

Coaches like Frank Carroll are extremely conscious of that fact. "Anything you can do in this sport to make yourself more beautiful, whether it's a pair of mesh hose, or a gorgeous costume, or a new hairstyle, or a fixed nose is going to have an impact on the judges.

"In this sport, if a girl doesn't have the right body, the right weight, the right looks, she's not going to make it because it's all based on beauty. I don't remember whether the operation was my suggestion but I know that it was always in the back of my mind, even when Linda was a little kid."

As a youngster (she started skating at the age of 9), Fratianne "really liked to show off. It was like, 'Hey, I can skate faster than you,' she recalled.

Somewhere along the way, she lost the desire to show off. "Even after she started winning competitions," said her mother, Virginia, "Frank would have to say, 'Now, get out there and take a bow. You don't have to back out of everybody's way.'"

"It's not like I had a mental block against it (showing off)," Fratianne said. "But I was a little self conscious. It was like if I saw a picture of myself, I'd say, 'Oh, God, look at my nose.'

"I was pretty shy and whithin myself," she continued. "Now I'm not as timid, I feel more confident on the ice. I feel I can look at the audience and smile and not think, 'Oh, God, what are they thinking of my nose?"

It is easier to skate like a champion when you know that people think you look like one. Or, as Frank Carroll put it, "You have to feel pretty to feel that, people are going to think you are." Carroll says there "has been a world of difference in Fratianne's skating since the surgery. She's using her body more. She's sexier. She looks up and makes contact with the audience."

To improve her performance value, Carroll schooled her in the art of spontaneity. She spent hours before a mirror practicing how to focus on someone in the audience and how to make her face light up. These days, she no longer bother to practice lighting up in front of mirrors.

In a sport where a coaches speak of producing a champion, Fratianne may have been overproduced. She admits she felt "a little bit like a robot." Now she must start pushing her own buttons.

Last year, while choreographing her program to the music from "Carmen," in which she skated the part of opera's most famous tart, Carroll turned to her and said, "Sorry, Toots, I'm not going to tell you what to do with your arm. You're the ballerina.

While Fratianne searches for the artist within herself, her coach is searching for new music to replace "Carmen," which some felt was a bit old for her, anyway. "Brooke Shields, she's not," is how one expert put it.

The new Olympic program is likely to be 'All-Armerican," said Carroll. "She's not going to try to anybody but the All-American girl, which she has not done.

Maybe then being Linda Fratianne will be good enough on and off the ice. CAPTION: Picture, "Skating is . . . like a beauty pageant," says Linda Fratianne.