Any attorney can file any sort of lawsuit in front of a court that he is admitted to practice. Class Actions are usually federal matters so that makes it pretty open. However, in order to file a class action, you do have to run down members of the class, etc... and that's not something a guy hanging out his shingle next to the county courthouse is going to have the resources to do.
Usually, these suits are brought by firms that either have class action experience, or they have some specific experience with the subject matter of the lawsuit. For example, a company that has broadcast radio/TV experience might work on a class action involving TV advertising or the like.