What can you get from stability shoes?

Pronation control. Over pronators can get the right running support from stability shoes, since it reduces over pronation and keeps the gait straight during the transition. A stability platform is support focused, but it does not control your foot motion too much.
Premium cushioning. Since stability shoes are supportive, you can enjoy a plush underfoot feel. Aside from comfort, the cushioning also softens the impact on the road or trail, and reduces pressure in the leg and foot joints.
Arch Support. The arch support in stability shoes is more prominent to support semi-low arches. It keeps the foot from rolling inward during the gait cycle.
Durability. Thanks to the thick cushioning and premium midsole materials, the durability threshold of stability pairs are higher than neutral or minimalist ones.

What to consider when looking for?

Trail or track? Before picking a stability pair, consider if you are comfortable on the trail or you want to stick to the pavement. There are stability shoes that are made for both, which is a better investment in the long run.
Midsole cushioning. The midsole materials should be on top of your priority list, since this would define how much pronation control your shoe can provide. Look for thicker cushioning, especially if you are a heavyset runner, so the plushness can properly support your running movement.
Shanks. A shank is usually found in the midfoot area. Made from TPU and other flexible plastic, you should make sure that the material is comfortable when you are running on it. For runners with a mild over pronation, a flexible shank would do.

Support and cushioning. The best stability shoes feature a great combination of lateral and medial support and firmer midsole. It is important to take time finding a shoe that have all these features packed in one.
Control features. The right stability shoes have impressive control features. Most likely they have rigid heel counter to secure the heel and reduce overpronation.
Durability. Reliable stability shoes have impressive durability feature for long-lasting use. If the shoe wears out quickly then, definitely it is not the perfect running shoe for you.