NEW YORK, Oct 30 - The number of help wanted ads in major U.S. newspapers was unchanged in September, a report said on Thursday, indicating the labor market may be stabilizing. The Help-Wanted Advertising Index, a barometer of the availability of new jobs, held steady at 37 in September, according to private research firm the Conference Board. "While employment gains have been scant, the number of layoffs has slowed," said Ken Goldstein, an economist at the Conference Board. The stabilization of job ads comes just after a government report published on Thursday showed first-time claims for unemployment benefits fell by 5,000 to 386,000. The Conference Board compiles its index by surveying help-wanted ad volume at 51 newspapers across the country each month.||