Running

Sounds flippant answer is depends on when they wear out. If they hurt to run in and are falling apart defo replace.

Manufacturer's answer: 200-300 miles for race shoes; 500 miles for training shoes.

In reality, all sorts of variables including how heavy/heavy-footed a runner you are, which shoes, how hard-wearing the outsole/midsole cushioning... My preference is to keep wearing a pair of shoes if there is still some rubber on the outsole and you're not running on midsole foam, and the shoes aren't otherwise falling apart. That's usually 1200 -1500 miles and has applied to different models from Nike, Adidas and Brooks. I've currently got two pairs of Adidas Glide Boost on the go, one at 1400 miles I now just use for easy runs on grass, the newer ones (400 miles) for most other things. The newer pair maybe feel a little more springy but there's not much in it.

To check for mid sole wear/compression, remove the insole and feel for notable depressions where your heel and forefoot bones are. With a finger inside and on the outside of the sole, you should be able to gauge how compressed they are.