Starring Frances McDormand. Rated 14A

Picture an Andy Griffith Show in which that little town is wracked by unsolved sex-crimes, Sheriff Andy has terminal cancer, and Barney Fife is an alcoholic racist with mommy issues, and you’ve got some idea what goes on in Ebbing, Missouri.

As in UK playwright-turned writer-director Martin McDonagh’s first two movies (In Bruges and Seven Psychopaths), Three Billboards is a wildly entertaining exercise in stylistic pastiche. It’s easily his best, both for multilayered storytelling ambition and as a vehicle for Frances McDormand, in her most expansive role since Fargo, as Mildred Hayes, a foul-mouthed Rosie the Riveter for our post-industrial age.

Mildred’s little Missouri town (actually North Carolina) is pretty much post-everything, especially since her teenage daughter was raped and murdered there. No clue has turned up in more than seven months, and she blames Sheriff Willoughby (a remarkably subtle Woody Harrelson) for slacking off. That’s where the title comes in, because she hires a local entrepreneur (Caleb Landry Jones) to slap up accusations that get crafty Ebbengites up in arms.

McDonagh switches effortlessly between outright comedy, sudden violence, and alarming tenderness. For example, the initially crusty Willoughby, facing his own mortality, turns out to have surprisingly Andy-like compassion. The bigger problem is his overly protective deputy, Sam Rockwell’s flickering-bulb Jason Dixon, whose very name is a reminder of the state’s proximity to slavery (across the Mason-Dixon line). “That’s persons of colour-torturing police,” he shoots back when Mildred accuses him with the N-word version.

Also in the mix are the “town midget” (Peter Dinklage does occasionally correct them), Mildred’s Opie-like teen son (Manchester by the Sea’s Lucas Hedges), and her brutal ex-husband (John Hawkes), now living with a dumb young girlfriend (funny Australian Samara Weaving).

With its reams of Tarantinoesque dialogue about sexism and racism, tinged by Coen brothers social commentary, Mildred turns out to the only fully-formed female on hand, with others either peripheral, ditzy, drunk, or incongruously pure, like Willoughby’s wife (fellow Aussie Abbie Cornish, using who-knows-what accent). A black detective (The Wire’s Clarke Peters) is introduced eventually, but the script doesn’t know what to do with him.

Like the movie itself, Ebbing is amazingly forgiving of increasingly outlandish outbursts and twists. But then, we loved Mayberry not for its realism but for its dream-like state of off-kilter grace.

Here, that grace has an edge. And it cuts.