Dancers can benefit from adding a little bit of light jogging into their training regime. In addition to boosting cardiac output, running helps to strengthen the stabilizing muscles of the legs and ankles. But to get into the hobby of running safely, you'll need to spend a little bit of cash. But putting that money into a Nike Lunar Glide 8 will last you a good number of years. (No, Nike doesn't pay me to say this.)
At the time I'm writing this product review, my running shoes are about 2 years old with about 400 miles on them. They've been through puddles, smothered by volcanic ash, up the Rocky Steps in Philadelphia, and through redwood forests in the Pacific Northwest. They've been crammed into tiny travel backpacks, crushed underneath piles of junk, and covered in glitter (long story.) They've experienced just about everything an outdoor athletic shoe is supposed to, and a bunch more stuff that they're not supposed to. And I still wear them.
I've only been caught out in a summer storm twice. Each time, the soles of the shoes were soaked thoroughly until disgusting squish sounds accompanied every step. The insole comes out easily for quick drying, and the material retains it's soft, spongy, shock absorbing qualities after drying. Additionally, the shoe maintains it's shape. Shoes with leather or any kind of canvas material will absolutely deform in the rain.
I don't wear them in the dance studio. The shoes are designed to withstand very specific pressures that your foot experiences when you run - but NOT necessarily the pressures when you dance. Running shoes don't hold up too well with quick lateral movements, which you will find in many urban dance styles. Putting these kinds of pressure on the shoes may cause the stitching to come loose.
This unpaid advertisement is all to say that investing in a good pair of running shoes will serve you well in the long run. Every athletics store has a densely packed wall of running shoes, and I can't speak to the quality of 99% of them. But, I can say that my purchase in particular has been absolutely worth the value.