Raised heel or flat for running shoes?
Various running shoes have varying "drops" from heel to toe. Some say the more 'drop' between heel and toe, the higher risk of injury because it encourages heel striking (which has been attributed to some knee injuries). However, that may not be true. A study was done that had runners running in various 'drop' shoes, and the ultimate findings were that the injury rate was the same between running in a shoe with low/no drop and running in shoes that had up to a 10 mm drop. So basically, use what works for you and your feet/gait.
Flat for running shoes
They should be contoured inside to support your feet. You can't really run with Keds, for instance.