The Supreme Court has a Lemon Test (for church-and-state separation), a Miller Test (obscenity) and a Smith Test (religious freedom), not to mention the late Justice Potter Stewart’s pornography test: He knew it when he saw it.

To this list, Justice Elena Kagan proposed adding another method of jurisprudence — the Cockapoo Test. The newest justice’s idea goes something like this: If the police dog who is sniffing for drugs outside your house is extremely cute, the procedure may not violate the Fourth Amendment prohibition of unreasonable searches.

“Suppose this really were a very simple procedure,” Kagan told the lawyer who was arguing that drug-sniffing dogs cannot be used to justify search warrants. “The dog comes up, takes a sniff, barks, sits down. And … the dog is not a scary-looking dog; the dog is a cockapoo. So just like, you know, your neighbor with his cockapoo, walks up to your door all the time, that’s what this police officer has done.”

The lawyer, Howard K. Blumberg, conceded that the dog in the case, Franky, although not a cockapoo, “appears to be a very cute dog.” But, he argued, “it’s not what the dog looks like, it’s what the dog is doing on the front porch.”

The Cockapoo Test, alas, may not be one for the casebooks. But after Wednesday morning at the Supreme Court, this much we know for sure: The Roberts court has gone to the dogs.

The justices spent two hours hearing arguments about drug-sniffing dogs, and whether the evidence their wet noses detect outside of homes and cars can be used to justify searches. Dog theory is a little-explored area of law, and the justices were ready to chew on it.

“I’m not sure it’s relevant,” Chief Justice John Roberts said, but “if you have a dog that’s trained and good at sniffing out heroin, the same dog is going to be good at detecting a bomb?”

“I think any dog could be trained in either discipline,” answered Justice Department lawyer Joseph Palmore.

“Can they be good at bombs, but not good at meth?” the chief justice persisted.

“I think once a dog kind of chooses a major, that’s what they stick with,” Palmore posited.

“You don’t want coon dogs chasing squirrels,” agreed Justice Antonin Scalia.

The canine court’s theme for the day unleashed all manner of wordplay. The Supreme Court reporters traded prospective puns before the argument, about marking territory and throwing bones. But there was a serious question: Justice may be blind, but can she admit as evidence a dog’s superior olfactory powers?

“There is an implied consent for people, visitors, salesmen, Girl Scouts, trick-or-treaters, to come up to your house and knock on the door,” argued Gregory Garre, representing the state of Florida, which is pro-sniffing-dog.

“Yes, there’s an implied invitation to the Girl Scout cookie seller, to the postman, even to the police officer, but not police officer with dog,” said Justice Ruth Bader Ginsburg.

“If you’re allergic to animals, you don’t want dogs walking around at your door,” contributed Justice Sonia Sotomayor, who also spoke about the importance of leashes.

“Well,” suggested Garre, “you can certainly put the ‘No Dogs Allowed’ sign out front.”

But the pro-dog argument was meeting with distemper on the bench.

Justice Stephen Breyer said a typical homeowner would resent man and beast coming to their door and “not knocking . . . just sniffing.”

Kagan questioned the logic of regarding Franky the dog not as a technology but as “just like a guy.”

“Franky is using the same sense of smell that dogs have used for centuries,” Garre argued, informing the justices that “Scotland Yard used dogs to track Jack the Ripper.”

The argument devolved into a dog’s breakfast, covering such legal topics as: the use of mothballs to mask the smell of marijuana; the smoke that comes from burning a corpse; delivery of invitations to the Policeman’s Ball; and how a tennis ball in a car trunk can cause a false-positive sniff for drugs.

In the second case, the nation’s brightest legal minds found themselves deep in the details of dog training and certification — an irony observed by Scalia. “That’s a constitutional requirement, that the dog training doesn’t count unless it’s training with the officer who is using the dog?” he asked.

“At least,” Scalia observed a few minutes later, “we don’t have to worry about mothballs in this case.”