What's the best kind a running show for someone that's pretty much flat-footed?

yikes. you should go to a pediatrist and get custom-made inserts for your shoes. My mom and sister have VERY flat feet and they now have serious ankle, knee and foot problems because they didnt do this soon enough. I doubt you will find a commercial shoe with enough support - depending on how flat your foot is - you will need to find additional support if possible.

Instead of going to a mall or general athletics store, I recommend going to a running/sports specialty store where they can look at your stride and how your foot hits the ground. The people who work there are usually very experienced/long-time athletes who can recommend a shoe that will fit your needs.

Your instructor used to be flawed. Your frame used to be supposed to run barefoot, however the excellent common touchdown for any distance longer than a dash is at the ball of your foot. When you are sprinting, you will have to land to your feet. Landing is those locations will make you a quicker, extra effective, and no more damage susceptible runner.

for this problem i suggest you avoid asics and new balance. go to under Armour's new line of shoes i forget the name of them. there are two types one its neutral and the other is stability. get the stability ones they work phenomenally.