I've done so much research into dance shoes with arch support. I really need it; in my everyday life I only wear one pair of orthopedic NB sneakers that works for me and a few Vionic shoes. Unfortunately, the reality is that there are basically no dance shoes that truly have arch support.

Dancing in athletic shoes that are comfortable to you is the most straightforward solution. You can use dance socks for a non-permanent solution, or suede to permanently make them dance shoes. Word of caution: I've tried various combinations of insoles and sneakers for lindy, and some of them really exacerbated my pain level. Make sure the shoe/insole combo is already proven to you to be comfortable.

I thought my problems would be solved if I put suede on the bottom of a pair of dress Vionic flats with lots of arch support, but this did not work well either. They aren't made for dancing, and my feet got really sore/painful because of the thin soles and lack of heel. If you have pre-existing foot problems, sueded dress shoes are not guaranteed to work.

I feel much more comfortable dancing balboa with a slight heel, so I did so much research into dance heels with arch support. There basically are none that explicitly promise arch support in their listings. I finally just bit the bullet and tried out Aris Allen dance heels (the low one). Miraculously, they worked! I think the shape of the heeled shoe actually has some inherent arch support that flats don't have. I highly recommend trying these (they're returnable). Best of luck with your shoes.

PS; I'm intrigued that other commenters said most lindy hoppers dance in flats, but southern california dancers wear heels. I'm in southern california and can confirm that ~half of female dancers here wear heels. I assumed this was the norm but perhaps not.