Running shoes are matched to the shape and size of the person's feet. If you have high arch feet, a good brand would be Asics, cumulus, or nimbus and if these don't have an adequate arch you can always add a gel insole.
Yes, you can purchase some good running shoes. Isoplus have real good running shoes and they support your foot so that you can get a good arch in your feet.
Typically, they are seen as a good brand, though there is some disagreement on the matter. Shoes are very personal, and as such will never be good for everyone.
There are several brands that offer good running shoes. Nike is a good brand.
The New Balance Minimus Life shoes are designed so the runner feels like he/she is running barefoot. Because of this, it is lightweight but has not arch support for over-pronators.
The Nike Shox brand is the best brand of lightweight running shoes. They cost over $100, but are very good shoes.
Axis primarily makes skate shoes, so they are not recommended for running. Nike or a similar athletic brand would be more suitable for running shoes.
The reviews on the Asic's Kayano running shoes are mostly good. They are made with a higher arch and a lot of foot support. If you do not have a high arch I would not recommend them.
Skechers aren't very good at promoting their running shoes. They are better at general tennis or athletic shoes. I would check out a different brand for running.
I got Asics gel Kayano running shoes for my brother and he loved it. He said they were comfortable and soft, so judging from his reaction, I believe the Asics gel Kayano running shoes are a good brand.
I wouldn't know if minimalist running shoes are good. I haven't tried this brand. You would have to go online or ask someone who has tried this type of brand.
There are many brands, and retailers you can buy running shoes from. You have to specify a brand you are interested in so that we can specify where to shop.
Generally more cushioned shoes are better for those with higher arches, you may need something with less cushioning to help the roll of your foot. Stability shoes with support would be a good place to start looking for your new running shoes.