Most undocumented immigrants at some point during their illegal stay in the United States of America have used a fake or stolen SS# to do normal things like going to school/college, banking, employment and filing taxes. If you are able to legalize your status either using one of the means available to adjust status through marriage or with some sort of amnesty, you will have a valid SSN.
So what will happen to all the paper trail and records created with a fraudulent Social Security number? Unfortunately, even if you leave your current job (where it is impossible to update the number) and start a new job, the record is not going away anywhere. For example, transcripts cannot be updated with new SSN (you just can't show up at a school or college years later and insist that they issue new ones to you with a different SS#). Going to a financial institution and trying to update the Social Security number might actually lead to a criminal investigation by the anti-fraud unit and filing of charges like in the case of Joe Giudice. The worst, or course, are the all powerful credit bureaus which have notoriously inaccurate records even for native born Americans. These companies have zero interest in making their records accurate or up to date and are known for harassing anyone who tries to approach them even when their are clearly wrong. So for you, the credit bureaus will gladly take your new information and add it to their databases, but the old information will never go away and will always be part of your credit history. The net result is that your credit file will always show your past use of stolen numbers because the bureaus use several identifiers (like phone numbers, addresses, credit accounts held, etc.) rather than just SSN.
Is it possible to make a clean start? To start off, try to change your SSN wherever possible. For example, just get a new job. Try closing your credit cards or car loans by paying down the debt in full. As long as you have a good record of paying down your debt when it becomes and have the income to qualify for a loan at this time, most likely the bank will not pay much attention to identifiers that the bureaus used. Time eventually erases everything. After several years of using credit responsibly, leading a crime free life, and being a good citizen, you can move forward in life in America.