Bill Bradley says he wants to register every pistol, just like every car. Vice President Gore's proposal restricting handguns would require potential purchasers to show a photo identification card proving they had passed a background check but would not affect weapons people already own.

With different plans but a common desire to avoid going too far, the two rivals for the Democratic presidential nomination are giving the sensitive question of limiting gun ownership the most prominence it has received in national politics.

Gore and Bradley are backed by 25 years of polls showing a large majority of voters support such measures. And they are tapping into the new fear that the televised aftermath of shootings in schools and workplaces has brought to suburbanites who had viewed guns as a problem for places where the walls have graffiti and the windows have bars.

The calculus for each candidate is how to craft a plan strong enough to attract the liberal Democrats needed to secure the nomination without alienating general election voters.

This afternoon, Bradley brought his campaign for gun control to one of the suburbs both candidates covet. Shaker Heights, an integrated and largely well-heeled city of 31,000 just east of Cleveland, is mostly Democratic and relatively liberal.

He spoke within sight of the corner where last March, a freshman at Shaker Heights High School was shot to death as she walked to class by a 21-year-old college dropout who had been stalking her. Suddenly, this peaceful place was filled with questions about how the shy man, who had been a psychiatric patient and later pleaded guilty, had gotten his 9mm semiautomatic pistol.

"It seems to be easier to get a gun these days than it is for me to get size 10 shoes--everyone wears 'em, so the store's always out of 'em," said Lisa Feeling, 37, a lawyer here who stays home with her daughters, ages 6, 8 and 11. "If you put up some of these little bars, getting a gun at least would require a little more creativity."

A month after the Shaker Heights freshman was shot four times, Eric Harris and Dylan Klebold opened fire at Columbine High School in Littleton, Colo., killing a teacher, 12 students and themselves. Although crime rates are going down, the psychic impact of Columbine has led many people who follow gun control to believe that 2000 will be the year that it becomes a decisive issue for some candidate.

"Gun violence used to mean a mother in a housing project keeping her kids away from the windows, but now it means kids in wealthy areas being required to use transparent book-bags," said Philip J. Cook, a Duke University economist who has studied gun violence for 20 years. "Guns are now perceived as a direct threat to the middle class."

Brad Coker, managing director of Mason-Dixon Polling and Research Inc., said the gun plans would be most helpful for Gore and Bradley with the young working mothers who were a crucial element of President Clinton's coalition. "Gun control may be a trump card in the battle for that group of voters," Coker said. "And we're always one incident away from the issue vaulting front and center."

When Bradley announced what he often calls "the boldest proposal for gun control of any candidate in the history of presidential politics," he leavened his prescription with a warm reminiscence about the days when he and his grandfather used a .22-caliber rifle to shoot targets down by the Mississippi River. Bradley went on to say he favored a registration system for handguns, a ban on Saturday-night specials and other "junk guns," and a national law limiting each buyer to one handgun a month. He since has bashed the vice president's alternative as "too timid."

Under Gore's proposal, federal law would require each state to set up a photo-licensing system for potential gun purchasers, with the minimum requirement that they had passed a background check and had demonstrated knowledge of gun safety rules. Gore, whose father was among the first southern senators to favor gun control, also wants to ban Saturday-night specials.

The National Rifle Association sees only semantic differences between the two plans. James J. Baker, the group's chief lobbyist, called Gore's proposal an effort "to license people who want to exercise their constitutional rights," and said Bradley's registry would simply produce "a list of firearms that have never been used in crimes."

There clearly is a audience for the ideas, however. In a Washington Post poll last month, 41 percent of those surveyed said not enough was being done to make it harder to own a gun--a larger number than were worried about jobs, traffic or taxes. This summer, 75 percent of those surveyed as part of a Washington Post-ABC News Poll said they favored requiring handgun owners to register with the government. And the Gallup Poll has found that going back to 1975, a fairly consistent two-thirds or more of those polled have favored registration of all firearms.

Frank Luntz, a Republican pollster, argues that those numbers are fool's gold for Democrats. "Since Columbine, support for new gun legislation has barely increased, but concern about moral decline in the country has shot up. People are more concerned about the root causes of crime, like the breakup of the family, and when you give them alternatives to gun control, they choose those."

For that reason, Herb Asher, a political scientist at Ohio State University, said the Democrats would be wise--especially in the general election--to promote their gun control ideas as part of a package of school safety and public health issues. "Gun control has become a surrogate for which party has been captured by which extreme wing," Asher said.

On the GOP side, Elizabeth Dole made gun control a major issue before she dropped out of the race. She clutched a trigger lock while she gave the commencement address at Yale in May, and endured boos from party regulars as she announced her stands. But neither of the two leading Republicans, George W. Bush and John McCain, has proposed a dramatic gun control measure. Bush has said, "The best way to protect our citizens is to vigorously enforce the tough laws we have on the books."

More than 500 people jammed the community center here to listen to Bradley, and the police turned away 100 more. Bradley quickly turned to gun control, saying, "I think it's about time somebody stood up and took on the NRA in a national campaign."

But Bradley acknowledged that winning passage of his agenda would be a huge task. He said he hoped to win a mandate like Ronald Reagan's in 1980, which was so huge many Democrats supported his programs. "If I win by a big margin in a general election, it will have the same effect on Republicans, because I think a lot of Republicans agree with what I'm saying."

CAPTION: Bill Bradley outlined his gun control proposals in a speech in the Cleveland suburb of Shaker Heights.