Being allergic to dogs and/or cats is a cause of unhappiness. It’s not uncommon in a family of, say, five for one kid to be allergic, so none of the other kids can have a dog or cat; thus, the interest in supposedly hypoallergenic breeds.
From the Washington Post:

Hypoallergenic pets may be only a myth, according to a study of 60 dog breeds
By Carolyn Butler, Tuesday, August 2, 1:33 AM

I’ve been suspicious of all so-called hypoallergenic pets ever since my husband first came face to face with his parents’ ragdoll cat, Posey - an adorable fluffball of a kitten who, the breeder improbably guaranteed, would neither shed nor cause allergic symptoms. He took one look and promptly started sniffling and sneezing.

There has been very little hard research on the topic, even as the market for supposedly allergy-free animals - which often sell for hundreds, if not thousands, of dollars - has boomed. (Even the White House succumbed to the trend , with First Pooch Bo, a Portuguese water dog who was chosen because of Malia Obama’s allergies.)

But a study in the American Journal of Rhinology & Allergy suggests that there may be no such thing as a hypoallergenic canine, after all.

Researchers at Henry Ford Hospital in Detroit analyzed dust samples from 173 dog-owning households, representing 60 breeds, including 11 that are considered hypoallergenic, including Portuguese water dogs, poodles and schnauzers. They found that the homes with allegedly hypoallergenic pets contained just as much of the prime dog allergen, known as Can f 1, as those with the other breeds. “Any way we looked at it, there just wasn’t a difference,” says senior author and epidemiologist Christine Cole Johnson. “There is simply no environmental evidence that any particular dog breed produces more or less allergen in the home than another one.”

… That’s not to say, however, that every animal generates the same quantity of dander. “The bottom line is that there’s huge variability from one dog to another in the amount of allergen they produce, but that variability is not predicted by breed, size, shedding or hair length - any of the things we thought in the past or that breeders still claim,” says Robert Wood, director of pediatric allergy and immunology at the Johns Hopkins Children’s Center in Baltimore. In fact, Wood notes that it’s not uncommon, within a single breed, to see a hundredfold difference in the amount of Can f 1 one dog creates vs. another. He attributes this to a combination of genetics and behavior as well as environmental factors such as how often owners clean their pets and their home. Still, generally speaking, Wood says that male animals tend to produce and shed more allergens than females.

Unfortunately, there’s no way to know how one bichon frise or German shepherd stacks up against another, allergen-wise, when you pick out a puppy. The only real solution, it seems, is trial and error.

But it’s hard to buy a puppy, then take him back a week later because you are allergic to him.
If there really are 100X differences between individual dogs, but not between breeds on average, then, presumably, artificial selection of existing breeds wasn’t aimed at allergenic properties, but at behaviors and looks. (Perhaps people are more prone to allergies today than in the 19th Century heyday of creating new breeds.)
So, it shouldn’t be that hard to create a truly hypoallergenic new breed. If they knew in Victorian times what we know now about the science of allergies, people in the days of Darwin and Galton would have come up with new breeds to do the job.
But, creating new breeds isn’t terribly popular anymore. We live in an era of great traditionalism about canine biodiversity.

I suspect that creating new functional breeds works better marketing-wise when the genes being selected for behavior pleiotropicly overlap with genes for looks. Breeds are their own advertising logos. Of course, when humans get overly obsessed with breeding for looks, they can lose the some of the functionality of a breed. But, there is an advantage to having a standardized look: if you want a dog that rescues people from drowning, you go buy a dog that looks like a Newfoundland.

Perhaps, the genes for being hypoallergenic don’t have much to do with how a dog looks.