If all the same price, then:
If you have a GSync monitor, go with the GTX 1060. If you have a FreeSync monitor, you should lean toward the RX 580, however, Nvidia's newer drivers might allow adaptive refresh on your monitor.
If you do go with AMD, I think the Nitro+ cooling solution is better, and the quick-change fans are convenient should you ever have to replace them. I think the Nitro+ is also clocked slightly faster than the Red Dragon.
The GTX 1060 is more power-efficient, and worth considering since all three cards are the same price.