The Central Intelligence Agency of “Spy” is maybe the least competent and most ridiculous CIA in movie history.

A war room in the basement of Langley headquarters is infested with mice and bats, and the agents working the computers and acting as eyes and ears for the spies in the field seem more concerned with office birthday parties than, you know, making sure nobody on their team gets killed out there.

In the field, many of the CIA operatives are either corrupt or bungling or both.

As one amused arms dealer puts it: “What is happening at the CIA? Do the drones have all the good jobs now?”

“Spy” is a foul-mouthed, often hilariously disgusting, slightly padded comedy that soars on the strengths of writer-director Paul Feig’s wonderfully idiotic script and nimble camerawork, and the bountiful comedic talents of Melissa McCarthy, Rose Byrne and Jason Statham.

Yes, Jason Statham.

This is the third collaboration between Feig and McCarthy, after “Bridesmaids” and “The Heat.” They’re three for three.

There’s not much of a plot here, just some standard spy-movie stuff about a villainess with a nuclear weapon and the race by the good guys to stop her before she sells it to the highest bidder. (One cringe-inducing but sharply funny line: “This nuke will be dropped in New York City by next week. So if you haven’t seen ‘Phantom’ ... ”)

McCarthy plays Susan Cooper, a nearly invisible agent who has spent 10 years at her computer, guiding the Bond-like Agent Bradley Fine (an excellent Jude Law) as he navigates the dangerous waters of international cool spy guy stuff. (They communicate through an earpiece he wears and a contact lens camera.)

Fine wears a tuxedo like a second skin and engages in snappy banter while disposing of a half-dozen henchmen at a time. Coop, clearly smitten with Fine, gets pats on the back for her excellent research skills and the baked goods she brings in on a regular basis.

But then something terrible happens, and most of the agents in the field have been compromised, so Coop volunteers to go undercover in Paris to gather as much information as she can and report back.

And with that somewhat clunky flick of the switch, Coop morphs from a timid wallflower into a trash-talking badass who packs a mean punch, masters various disguises ranging from Single Mom in a Bad Christmas Sweater to Midwestern Mom With a Doll Collection. (“Why don’t I just marry one of my dolls so I’ll be even sadder?” she laments.)

When we first see Jason Statham as the fierce and intense Agent Rick Ford (How many Brits are working for the CIA, anyway?), we figure it’s going to be the usual Statham role – the super-cool killing machine with a hundred tricks up his sleeve. Instead, Ford turns out to be an utterly unhinged anti-hero who spins increasingly bizarre tales of courage and suffering (“This arm was completely ripped off, and I used this arm to put it back on!”) while creating more chaos with his clumsiness than any law enforcement figure since Inspector Clouseau. He’s a fantastic moron. At one point, he suggests going undercover via the “Face/Off” method. His supervisor has to explain that was a movie and not something they can actually do.

Byrne shines as Rayna Boyanov, who has the nuke in her possession and routinely orders one of her hit men to kill another one if there’s a screwup. Rayna isn’t a great villainess – she’s too loopy to be intimidating – but McCarthy and her “Bridesmaids” co-star have terrific chemistry together. Whether it’s Rayna dissing Coop’s outfits and eating habits or Coop mocking Rayna’s backstory and ridiculous hair, it’s great verbal tennis.

As the action moves from Paris to Rome to Budapest, and we meet new characters ranging from the consistently funny (Peter Serafinowicz as an Italian spy constantly hitting on and groping Coop) to the not-as-funny-as-it-should-be (50 Cent as 50 Cent), “Spy” starts to wheeze a bit as it reaches the finish line. This is not a story that needs two hours to be told.

What does work, in every scene, is McCarthy’s performance. She’s as funny and as winning as anyone in the movies these days. Rated R.