With the exception of the ongoing high-profile cases of Joel Tannenbaum and Jammie Rasset, record labels have moved away from suing individuals for big money for copyright infringement. However, there may be a new approach to getting money from regular Joes downloading copyrighted material: There's a company called Digital Rights Corp. that goes out and finds suspected copyright infringers, and demands they pay $10 via a website. The approach, which promises fewer long court cases, and more effective ways of getting paid by labels, has been taken up by a couple of labels, but none are public at this point.
It should be noted that the legality of this approach of going after downloaders is in question, since the company basically just pressures people into paying without giving them easy to understand legal recourse. For more info on how this could wreck your world, head to Digital Music News.