She is rich and famous and the star of the most popular TV show in America. So what's troubling Roseanne Barr? Inquiring minds want to know.

So do the people in charge at ABC. They're worried. The network has a huge stake in the success of Roseanne - and in the stability of Roseanne.

Will Roseanne become the next Cosby, rolling along atop the Nielsen ratings for years, winning the hearts of viewers and earning, no exaggeration, tens of millions of dollars a year for its network?

Or will the hit show crash and burn like Moonlighting, collapsing under the weight of its stars' egos after losing key producers and suffering from turmoil, on the set and off?

Not too long ago, several ABC executives got an up-close-and-personal look at the tempestuous nature of Roseanne Barr.

They visited the set with the best of intentions. Theirs was a virtual pilgrimage, arranged to pay homage to the 36-year-old comedienne. Roseanne had just broken through to the No. 1 spot in the ratings, and the ABC brass knew by then that the star was sensitive. She would expect attention to be paid.

After some deliberation, the network honchos ordered an elaborate cake from the fanciest bakery in town and arranged to have it sent to the set before the evening taping. The idea was to honor not only Barr, but the cast, the crew and the writers for their work.

Barr, it turned out, was in no mood to celebrate. When the cake arrived, she exploded.

"You would give a cake to a fat woman!" she screamed, and proceeded to destroy it. One bystander said she toppled it. Another source, who heard the story secondhand, reported that she took a baseball bat to the cake.

Temper, temper.

Last month, though, the same ABC executives stood before a roomful of TV critics, brushed off questions about Barr and insisted that there is no cause for concern about the show.

Ted Harbert, the executive vice president of ABC Entertainment, said breezily: "Roseanne is looking forward to coming back to work and starting this season after she's done with her movie. We have no reason to think that everything's not going to be swell."

ABC executives said the same thing not long ago about Moonlighting, at a time when the feuding was intense among Cybill Shepherd, Bruce Willis and creator Glenn Gordon Caron.

One fact about Barr is inarguable: She has had a tumultuous year in the limelight.

Last summer at this time, Barr was a well-known stand-up comic who had appeared frequently on cable TV, but she was by no means a household name. She talked frankly then with TV critics about her life and her past - how she suffered prejudice as a Jewish girl growing up in Mormon-dominated Utah, how she overcame psychological problems as a teen-ager, how she lived for years with no money in a trailer with her postal-worker husband, how she struggled to survive in the brutal world of stand-up.

She was gratified, she said, to have the chance to do a TV series and to settle into a stable life in Encino, a Los Angeles suburb, with her husband of 14 years, Bill Pentland, and their children, Jessica, 13; Jennifer, 12, and Jake, 10.

"I feel happier inside myself now," she said. "I don't feel like life is such a big struggle."

Barr was loose and funny.

On watching TV: "I like to watch stuff that really makes me mad. I get lots of material from watching Donahue and Oprah. What else am I going to do at 3 in the afternoon? Exercise?"

On posing for Playboy:"I don't think I would do nude. Unless it was integral to the plot. Or somebody asked me."

Barr's new show had been created expressly for her stand-up persona, but it was also conceived as an ensemble piece. The title was originally Life and Stuff, which suited Matt Williams, the co-creator, head writer and executive producer of the show, who did not want the show to become a star vehicle.

But Barr wanted her name on the sitcom. Williams was supposed to be in charge, but he lost the argument over the title and then lost his job when he was forced off Roseanne midway through the first season.

Williams told the Los Angeles Times: "The problem was who was going to have the final say on stories, scripts and the overall thrust of the show. As the creator and executive producer, I automatically assumed, rightly or wrongly, that I would have the final say. And, obviously, Roseanne thought she had the final say. And that, in essence, was the central conflict."

Creative tensions, they call it in Hollywood. That means that Barr made so much trouble - losing her temper, yelling at cast members, refusing to come out of her dressing room and the like - that the executives who own the show, Marcy Carsey and Tom Werner, felt the best way to ease the tensions was to persuade Williams to step aside.

Carsey and Werner, promoting their new show Chicken Soup at a news conference last month, declined to discuss Roseanne, except to say that they thought the show was now in good hands.

Barr's personal life has also been stormy. Last spring, a magazine pried open the confidential records of a 15-year-old adoption proceeding and hounded Barr with questions about the child she gave up. She begged reporters not to pursue the story, but the teen-age girl was besieged by the press.

Last month, Barr filed for a divorce from Pentland. She is seeking custody of their three children and plans to marry Tom Arnold, a longtime friend, stand-up comic and writer on Roseanne.

The supermarket tabloids have been having a field day with Barr. Some recent headlines: "ROSEANNE'S WACKY WEDDING . . . And Elvis' bizarre role in ceremony"; "Socko catfight that could make or break Roseanne Barr . . . Meryl Streep slugs Roseanne in scene from She-Devil."

The tabloid stories are a price of fame, but they seem to be getting to Barr. She wrote a column for The New York Times, complaining that she is getting picked over by the press, and recently she appeared on ABC News' Prime Time Live to air her gripes.

"No matter if I was the richest, thinnest woman in the world, I would reserve the right to complain," Barr said. In her defense, Barr has been roughed up more than a little by the press.

Insiders say Barr, for all her problems, is committed to making the show work. Her supporters say she insists on control because she cares deeply about Roseanne.

"She would like very much to be recognized at some point, and have this program recognized as one of the true classics of network television," said Robert Iger, the new president of ABC Entertainment.