Do thigh-high stockings stay up on their own?

Some do. Stay-up styles have a grippy band, often silicone-lined, that holds them on your thigh. Others are designed to be worn with a garter belt or suspenders. Check the product description to see which type you're buying. For stay-ups, a correct thigh fit matters most, and skipping lotion on your thighs beforehand helps the band grip.