My daughter teaches at a Catholic high school. She owns a pair of "All-Rounders" by Mephisto, the French shoe manufacturer that makes comfortable footwear for walking, sport and dress. The All-Rounder is a cross-trainer that looks like a maryjane.
The school forbade her to wear them when she taught. The school says they are sneakers.
I have a pair as well. I wore mine frequently when I was a permanent sub at a high school and when working retail. Other women who see my shoes love them. They generally call them "dress sneakers." I wear mine with wool socks and find them so comfortable for a long day. One of the nice features is the strap is fastened with velcro and there is a velcro heel width adjustment, which means if standing all day makes your feet swell, you can ease give yourself a little more room in your shoes.