I'm sure many people, like myself, need running shoes for flat feet.
You have flat feet if you have a very low arch. Arches are supposed to be fairly high (though not too high - some people have arches that are too high) and flexible. Nature/evolution/God (whatever you choose to call it) designed our feet with an arch to better absorb the shock of the force of our feet hitting the ground with our weight at great speed, which magnifies the intensity -- up to eight times your body's weight. Our feet and ankles are marvels of practical engineering.
If your foot does not efficiently absorb this force, it's transmitted up your bones to your knee and hip joints and your back, and over time can damage them.
Part of the process of absorbing this force is that your foot rolls inward as it strikes the ground. This is called pronation.
Some pronation is natural and healthy. It's part of absorbing the shock of running, and is what your foot is designed to do, using your arch. And this is called neutral pronation, and about 20 to 30% of runners neutral pronators.
If your arch is flat, (or called low or fallen), then your foot tends to turn inward too much. This is called overpronation, and it tends to send too much force up your body, inbalancing the rest of your skeletal structure and muscles and ligaments.
I can't say that if your feet are flat that it's a certainty you overpronate, but it's certainly a high probability.
Conventional wisdom is that if you have flat feet, you need running shoes with built-in support -- much like the clunky shoes I wore as a kid.
The first level are what're called stability running shoes. These are designed to give your feet enough support and stability to prevent the overpronation, thus protecting the rest of your body.
Good ones include: Nike Zoom Structure Triax+ 14, Brooks Adrenaline GTS 9, New Balance 1226, Mizuno Wave Inspire 7, and Asics Kayano 17.
If these aren't enough, you can go for motion control running shoes. These include: Brooks Beast 10, Saucony Progrid Stabil CS, and Asics Evolution 4.
I can't comment on most of these. I use the Brooks Beast and have not had any problems. Your mileage may vary. (Most runners do put in more mileage than I do - I believe in running intervals, not long distance.)
Ironically, some experts tell those of us with flat feet to run barefoot (if safe) or in the new style "barefoot" shoes. We can also work on our lack of arch support by picking up ping pong balls with our toes.
I'm lucky enough that I can spend a lot of time walking around without shoes. Soon I'll be doing that wearing what I still call "thongs," but are now more widely known as flipflops. These are like being barefoot, but with some rubber to protect us from broken glass.
But for running, at least for now, I'll stick with my Brooks Beasts.
However, if I had to wear ordinary shoes again most of the day, I might try running in the new style of shoes that offer no support.
I personally believe that flat feet are caused by us not exercising our feet properly when we're very small. I know that my family were protective. I spent my first year of walking in a New York City apartment, and I suspect my mother kept me wearing shoes all the time.
Later, I lived with my grandfather who had a huge back yard full of grass and -- I'm sure -- no broken glass. But he would still have been afraid to let me go running outside in bare feet.
I did spend most of my childhood summers at a swimming club and so went around barefoot much of the time. But apparently this was not enough to offset the effect of wearing shoes from getting up to going to bed the rest of the year.
I remember being told as a kid I had flat feet but not really caring. I did have to wear heavy clunky leather shoes "for support," though I also wore regular old tennis shoes or sneakers when just playing. I don't recall anyone caring about my flat feet when I ran for my junior high track and cross country teams. But it also didn't even cross my mind that my feet should affect my running at all. Besides, I wasn't a very fast runner, and so didn't get much attention.
By the time I was grown up and became interested in running as a way to stay in shape without regular access to a swimming pool (as an athlete, I'm primarily a swimmer, but that's just not as convenient as running), I barely remembered that I had "flat feet." People joked about it being enough to keep you out of the Army, but I'd passed my draft physical (though was never called for duty), so I don't believe it was a real problem.
I suspect that in places where people normally let children run around in their bare feet, few if any of those kids have either flat feet or arches that are too high.
Therefore, I advise everybody who needs running shoes for flat feet to try out different kinds and brands and see which ones work the best for you.