Using a sports bra when running is a good idea for women who are well endowed up front and even for those with smaller chests if they run long distances and want to avoid issues with chafing. You want to be not only an explosive runner, but a comfortable one.
Breast tissue can is quite heavy and if it is unsupported or supported poorly it can put strain on your back. The best way to avoid issues is to make sure that your chest is properly supported and that you can move around freely with good compression and control of your chest.
Your breasts are naturally supported by collagen, but repeated strain and poor support will cause that collagen to be stressed and could cause damage. A poorly fitted bra could also cause damage, and if you try to run in a bra with underwiring it would be most uncomfortable. You may also find that common lingerie is quite clammy and that the way it clings to your skin would not work well with running.
Sports bras are made from a strong but stretchy material with panelled support in all the right areas. This means that you don’t need to worry about issues with excessive bouncing, chafing or rubbing. The weight is supported across your shoulders and back thanks to the thicker straps, and you can focus on your running form instead of the weight shifting around your chest.
Most women are still wearing bras that are the wrong size for them, so it’s important that you get your bra size checked before buying a new sports bra. Often, women wear too small a cup size and too large a back size (or vice versa) and both errors can cause your bra to feel uncomfortable and to pinch in the wrong places. A professional fitting service can help you determine the right size to wear.