I was told no rubbing alcohol
I remember my mom, aunts, and cousins using rubbing alcohol on their babies' umbilical cords but our pediatrician told us not to. He said that a study was done that showed baby's umbilical cords fell of a few days later then those who didn't. I don't think that's really a big deal, but we didn't use it just in case. We just kept it dry (sponge baths only) and made sure the diaper didn't cover it up so it stayed clean.
Avery's cord stub fell off when she was 5 days old. It was a bit gooey (gross!) right before it fell off and it smelled a bit. The nurse said that was super normal. A few days after it fell off we took a Q-tip with a bit of rubbing alcohol and really cleaned it out good. You can't be shy about it :)
We still clean out her belly button every once and a while. Oil can build up in those little things and get pretty scuzzy.