One thing you do want, in event shoes is to make sure they have a solid base so anything with an air bubble isn't ideal and preferably a shallow sole. The higher the heel part of the trainer the more of an unstable position this puts your ankle joint in. If you have a high or low arch it's important that you use soles that supports it.
If you develop flat foot secondary to you putting on weight it's ideal that you use an arch support as you can develop secondary knee issues as this can increase vagus force and can increase pressure through your medial femoral condyle and increase your rotational forces through your knee that can potentially cause meniscal problems and patella femoral joint problems. So if it is the case that your weight is causing your flat foot, you need your arch supported gradually so start low and gradually build up but don't go too high.
When you invest in insoles it a good tip to start with a soft arch and wear for 2-3 hour a day and gradually build up how often you wear it until your able to tolerate it for long periods and then build up to a firm arch.
It's a good idea if you have noticed you have flat feet or one more so than other to get it checked out by a physiotherapist, podiatrist specialising in biomechanics or doctor. As it can be due to weakness, or avulsion of you tibial is posterior muscle or nerve palsy causing inhibition of the muscle.
I typically wear low arch asics and they tend to support my arches and have a solid sole. Everyone has their preference. Some wear basketball shoes for the ankle support, some like converse as it gives them a little shock absorption but keep them in contact with the floor better. It's probably not advisable to wear anything with a high arch, soft lateral heel sections or a high heel section as this can put your ankle in an unstable position and at risk of ankle sprains particularly under high loads.
Make sure that what ever shoes you wear for event training and competition you keep them the same style for all. A big trick I learnt for all events is consistency, and when it comes to biomechanics if you change just one element it can alter how you lift, pull or push. For instance when i wear a heeled shoes like weightlifting shoes this increases the tension in my patella femoral, it puts my hamstrings and calves in a better position to contract, this also increases the lumbar arch because my pelvis is more anteriorly tilted. This in turn increases my ability to fix your core and also means you can you tend to extend through my thoracic to help you maintain an upright position. So using is as an example you can see the difference just altering the height of your heel in your shoes, however if i use trainers with too much of a heel this does flare up my tendonitis. But overall find what works for you try different types of shoe.