Bend and depress the heel box, or back third of the bottom of the shoe. It shouldn't bend easily. A sturdy heel box is better for any kind of exercise than a weak one because it offers you better cushioning and support.
Grab your foot with your hand. Bend it around and see how much your bones shift with the motions of your hands. If you have a rigid foot that doesn't move much when you manipulate it, you should buy a shoe that has a flexible top made of fabric or mesh. If you have a flexible foot that moves easily, find athletic shoes with sturdy tops that will keep your feet in place.
Check the arch of the shoe. This is the narrow area of the foot between the heel and the area after the toes. Buy a shoe that offers a wide arch, since it is more supportive than a shoe with a slender arch.
Step on a brown piece of paper while your foot is wet and look at what kind of curve exists between your heel and toes. If you have a slight or no curve, you want a shoe with motion control and lots of support. If you have a pronounced curve, find a shoe with a soft midsole and lots of cushioning. A neutral arch, which you may have if your shoes wear out evenly, means that you want a stability shoe that has both cushion and support.