I probably deserve to get shot for this, but it DOES work- even if I found it out on accident.
Okay- years ago, before I discovered a real stylist (a real hawtt one) and real Product, I used show-sheen on my hair in a pinch. I was spraying my hair upside-down, so when the mist hit the floor, it dried- I turned in it barefoot- and almost went down!
Anybody NOT from horse-ville doesn't know that Show-Sheen is a product made by Absorbine and is basically diluted, smelly, liquid silicone. It helps a horse's coat lay flat, sleek, and shiny... and is NOT NOT NOT for use in the area where the saddle goes- why? You'll slide RIGHT off!
Showsheen comes in a big spray bottle (trigger-nozzle), and when you spray it on your socks, you will slide, even on carpet. NOT recommended obviously for spraying on real dance shoes, but maybe a crappy pair or some socks. It MIGHT help- but probably won't. Best place is to just take lessons somewhere that cares about other people's legs!