A tonal disaster, Three Billboards Outside Ebbing, Missouri chronicles the angry campaign of a single mother (Frances McDormand) to get her small-town police department to continue investigating her daughter’s rape and murder after the trail has long gone cold. Doesn’t sound funny? That doesn’t keep writer-director Martin McDonagh (In Bruges) from going for laughs. McDormand’s dialogue mostly consists of vile, “shocking” insults that are delivered as punchlines; Peter Dinklage hangs around so characters can joke about his size; and Sam Rockwell yucks it up as a dopey local cop. That Rockwell’s character is also a violent racist and, much later, unconvincingly asked to be the story’s potential saving grace gives you an idea of the whiplash Three Billboards delivers. Throughout, comedy awkwardly bumps up against sentimental sincerity, often in the same scene and within the same character. (I thought Woody Harrelson, as the police chief, was supposed to be the funny-scary villain until the movie gave him cancer.) The relationships among all these characters are never fully established, so that watching Three Billboards feels a bit like being dropped into Season 3 of a sitcom without having seen a single episode. Ebbings registers less as a real place than a construct, a stage for some facile observations about justice and vengeance. “Anger begets anger,” a minor character notes at one point. Three Billboards Outside of Ebbing, Missouri giggles while its characters violently prove that maxim to be true, then lets itself off the hook with an ambiguous, “let the audience decide” ending. It’s as if the movie doesn’t really care enough about any of this to take its own stand.