“Three Billboards Outside Ebbing, Missouri,” the third feature written and directed by the playwright and filmmaker Martin McDonagh (“In Bruges,” “Seven Psychopaths”), sets itself up with no time to spare. A trio of old, weathered, falling-apart billboards stand in the morning fog, like broken dominoes, along a sparsely traveled road on the outskirts of Ebbing. Mildred Hayes (Frances McDormand), her face bent into a scowl of despair and resolve, marches into the advertising office that controls the billboards and draws up a contract to rent them for a year, offering a one-month down payment of $5,000. She then has the billboards painted red and adorned with three linked messages in black capital letters: “Still No Arrests?” “How Come, Chief Willoughby?” And finally, “Raped While Dying.”

Mildred, we learn, lost her teenage daughter, Angela, seventh months before, when the girl was raped, killed, and set ablaze (not necessarily in that order). The billboard messages are an attack on the complacency of the local police force, and especially its chief, Sheriff Bill Willoughby (Woody Harrelson), for having failed to find the killer, or to even put much effort into it. When Willoughby learns of the billboards, he blows a gasket, and it seems clear where the movie is headed: to a battle between the police and Mildred, the aggrieved citizen who has taken the law — or, at least, the power of public shame and coercion — into her own hands. When she accuses Willoughby of being “too busy torturing black folks” to solve her daughter’s murder, there’s an unmistakable echo of the case of the recently pardoned Arizona Sheriff Joe Arpaio, who in his anti-immigrant crackdown fever ran an office that failed to investigate hundreds of sex crimes against children.

Yet the black-and-white moral lines quickly bleed into shades of gray. Mildred, for one, turns out to be even more possessed than we thought. When Willoughby shows up to have a talk with her, he sounds, to our surprise, quite sincere in his desire to find the killer — and, what’s more, he reveals that he has cancer. Yet that does nothing to soften Mildred’s fixation on her billboards. (It just makes her say, “They won’t be as effective after you croak.”)

A local priest tries to calm her down, and she responds by likening the church to the Crips and the Bloods (“You’re culpable,” she says). A bit later, during an appointment with a dentist who’s friends with Willoughby, she grabs the live drill and plunges it down into his thumbnail, at which point it becomes clear that Mildred isn’t just pressuring the law — she’s vilifying what she views as a patriarchal conspiracy. She’s woke, she’s fierce, she’s beyond shame or scruples, she’s screaming truth to power, she’s charged up with the wrath of an avenger. But how far can she go? And to what end? Has she gone off her rocker with grief and rage? And where, exactly, is the movie going?

Not where you think. “Three Billboards Outside Ebbing, Missouri” isn’t a righteous demagogic attack on the complacency of the police, or on masculine violence and privilege — though it is a meditation on those things. It’s not a whodunit with a clear villain and a connect-the-dots suspense plot that will lead to his capture — though it plays off our desire for all that. Lastly, it’s not a tale that offers a pat resolution — though when it’s over, you feel like you’ve been on a journey, and that McDonagh has led us through the paces and pleasures of a three-act story in a stylized, postmodern way. In its vortex of agony and anger, forgiveness and redemption, “Three Billboards” may play, during awards season, as a kindred spirit to “Manchester by the Sea,” yet that movie was a masterpiece of dramatic realism. This one is more like a quirky emotional puzzle put together by a trickster poet. It’s far from a masterpiece, yet it holds you, it adds up, and it’s something to see.

It’s McDormand’s passion that welds the picture together. She makes Mildred a heroic fighter, but the actress has never played someone so eaten up by fury, to the point that it renders the character — and the performance — at once sympathetic and forbidding. Yet it’s Mildred’s glowering refusal to back down that defines her, and McDormand brilliantly spotlights the conflicted humanity beneath the stony façade. On some level Mildred is just a small-town single mom who works in a gift shop, but she wears her jumpsuit and gray-polka-dotted bandana like a soldier’s uniform, lashing out at everyone within earshot, a quest McDormand makes at once crazy-fearless, stubbornly infuriating, and noble. Mildred’s son, Robbie, is played by Lucas Hedges (from “Manchester by the Sea”) with a cautious poise that cues us to see that dealing with his mother has never been a picnic.

McDonagh, working with the cinematographer Ben Davis, gives the small-town country settings of “Three Billboards” a graphic visual spaciousness, yet unlike the Kenneth Lonergan of “Manchester,” he’s an example of how you can take the playwright out of the theater, but you can’t take the theater out of the playwright. It’s not that McDonagh’s dialogue is stagy; it’s tart and spontaneous. But he has built “Three Billboards” around the kind of constructed encounters, the carefully layered motifs and metaphors, that are the hallmark of a well-made play. Ebbing is a small town, but in this movie it’s the kind of small town that feels as if it has just nine people in it.

Yet each one makes an impact. Harrelson plays the hard-ass chief with more sorrow than anger, and John Hawkes, as Mildred’s ex-husband, has a sinewy arrogance that expresses the desperation he’s covering. Clarke Peters, as the new lawman in town, proves, once again, that he’s a remarkable actor, exuding a no-nonsense authority that’s like a tonic. And Sam Rockwell is a revelation. As Dixon, a racist cop who’s a loser, a momma’s boy, and a violent flake, he gives a high-wire performance, daring to make himself gnarly and dislikable, only to undergo a transformation that the actor, mining his moonstruck ability to win laughs in even the most disturbing situations, makes spiritually convincing.

As you watch “Three Billboards,” you can see the outline of a revenge film, yet it’s a kaleidoscope of a movie in which the emotions and alliances keep shifting. At one point, it leads us to believe that we’ve discovered the killer, but it also suggests that that would be too easy. By the end, it teases you with intimations of lone justice, only to draw back to something less reckless. Mostly, the movie takes the new war dance of feminine righteousness and masculine power and holds it up to the light in a nearly mythological way, asking the question, “Can’t they just get along?”