Let's take lulzsec as an example; they registered lulzsecurity.com. There are two problems that I don't understand how they solved:
- They had to pay for it. Tracking down money is generally much easier than tracking down IP addresses. I assume they didn't use stolen credit cards (with all the attention they received, people would have quickly found out and taken away their domain).. And even with prepaid credit cards it's relatively easy to find out who bought it, with security cameras/etc.
- They had to have played by ICANN's rules - again, because of the attention they received, if they hadn't people would have found out and they would have lost the domain. This means giving valid contact information.