In 2010 Fox News posted an article in which they quoted a representative of the FAA saying that the FAA might not regulate jetpacks (the representative did say that they could fall under Part 103). Do any regulations exist in the US for jetpacks?
This depends on what you mean by a "jetpack". If you mean the classic James Bond-style rocket pack, then according to an FAA spokesman quoted in the Wall Street Journal they just don't fly long enough for the FAA to be interested:
Once aloft, a jetpack pilot is preoccupied with getting down quickly: A typical pack holds about 10 gallons of fuel, only enough to fly for about half a minute. The Federal Aviation Administration doesn't regulate jetpacks. "Thirty seconds is not sufficient to be considered a flight," says FAA spokesman Les Dorr. He adds that it's up to the individual to assess the risks.
On the other hand if you're referring to the Martin Jetpack, which uses fans instead of rockets, then several sources (e.g. Wikipedia) say that it's considered an ultralight by the FAA, in which case all the usual ultralight regulations would apply. But I couldn't find any direct confirmation of that on Martin's site.