Most people wear regular skate or running shoes to mountain bike ride. However you should consider the type and shape of your pedals and the type of riding you are doing as this can have a serious impact on your feet.
Pedals that are clip-less may require special shoes that accept the particular cleat for your bike pedals. You will also want a stiff sole shoe to protect your feet against rough terrain.
Winter Mountain Bike Shoes
The feet are one of the extremities of your body which are most vulnerable to the cold. When you're not riding a bike your feet become cold quickly, but when you are riding the pedaling pressure can actually cut off toe blood circulation to a degree, bringing about the risk of frostbite if conditions are particularly chilly.
Special mountain bike shoes and booties are a must have for winter riding. The shoes should be well insulated and waterproof. The booties can zip over your shoes and many have a sole pattern specifically made for cleats.
Investing a suitable pair of mountain bike shoes is a wise choice.