Here in Holland they don't use them anymore, they use blue plastic clamps to clamp off the cords,I was thought by nurses to just fold the diaper edge under the cord and clamp and just leave it like that. With both my kids it fell off after 6 days. Both my kids, 3 years old and 4 month have normal bellybuttons, allthough the youngest one has somewhat of a ball in it that is still a bit swollen, but the doctor said that will go away eventually.
For the oldest one i did buy the bellybuttonbands, they where just yersey bands that you should close with a safety pin, but i never used those, because they said it was "oldfashioned".
But i know my sister and sisters in law used them on their kids and their youngest is 10 years older than my oldest so it must have changed somewhere in those 10 years