Probably no states will allow SSN for F2 dependents. There were few states which allowed SSN for F2 dependents for the purpose of Driving license. In the state of my residence, it was stopped in February 2002. I have seen few F2 dependents (who applied for certain professional licensure exam, passed it but couldn't get the licence without a SSN. They had to enroll to school, change from F2 to F1 and get the SSN. It's really tough these days.