I'm aware that there are spike shoes (for running) with permanent spikes. But often most that I've seen are rubber spikes.
So basically I was wondering is there such a thing as non-rubber permanent spike shoes?
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You are referring to Track and Field shoes, also known as Running Spikes. This will help in your searches online for the shoes. The running spikes are commonly made with metal spikes, as opposed to soccer shoes (or football as the rest of the world calls it) which tend to be rubber.
Running spikes are intended for short distance running, particularly sprinting on the type of surface you would have in track and field events. One (expensive) example would be the Nike Zoom Ja Track and Field shoes. You can also find many other examples at this "Nextag" link.
According to this link the spikes are replaceable so the manufacturer can sell one shoe and you can use the proper spike for your event. According to the article: