There is no real 'best' support as a single brand or product that will work for everyone. You need to first determine why your feet are sore, and find a product that will work to correct the underlying condition.
Puma makes great running shoes that provide excellent support for runners.
Running shoes provide comfort and support. If the shoes are old or worn they can no longer provide the proper support. It would be best to get new shoes every 100 miles.
Of all of the online stores that I have "visited", the Asics website has the best arch support running shoes. According to their website, you may find Asics online at www.footlocker.com, www.footaction.com, and www.sportsauthority.com.
Squatting in running shoes is generally safe, but it may not provide the best support or stability for heavy lifting. Consider using weightlifting shoes for better performance and safety during squats.
The reason why they are better than most shoes is because Mizuno is an upper end brand of running shoes and they have better support. But you always want the shoes that feel the best.
Asics Kayanos are the ultimate best running shoes out there that are easy on your budget and on your feet. They are lighter on your feet than a typical running shoe and offer a better support.
It depends on the type of running shoes that you like best. These are light weight, support high arches, and they are easy for those with a longer running stride.
A shoe with good arch and instep support is recommended for running and jogging. Many brands of shoes by Nike, Adidas, and Reebok have shoes for running, walking, and cross-training which would be suitable.
To the best of my knowledge the advantage of the Asics Gel running shoe is that it has gel in the insoles for more comfort and support. These shoes are available at Kohl's, Zappos, Amazon and Ebay.
If your looking to find a pair of running shoes online I would start with looking at runningwarehouse.com they have a very large variety of shoes to pick from that fit most every shoe size, and support type.
Cross trainers are probably going to be your best bet, since you won't need the specific support running shoes offer during an aerobics class.
Nike makes great running shoes that are somewhat affordable. You could even try paylessshosource.com for a cheaper option, just be sure any running shoe you buy has proper arch support.