So let us first talk about having a false SS# with respect to United States laws, whether it is completely phony (that is a number pulled out of a hat by crooks who sell forged documents online or in immigrant neighborhoods) or belongs to dead individual or a child or belongs to another individual who has either allowed you to use that number (you can not legally allow someone to use your SSN but you can definitely say that you will not report that as a crime) or you have simply decided to use that number and this person has not yet realized that he is a victim of identity theft. Because laws are tricky and complex, what you may want to know is that identity theft in general is a felony, but the courts have sometimes interpreted it to say that if the victim is not harmed, the criminal need not be punished too harshly. Having said that, as a matter of law, using a Social Security number that has not been legally assigned to you by the United States Social Security Administration is considered a criminal action. Similarly, abuse of the number, for example, working without authorization, is also criminal.
But what about bank accounts, credit cards, car/home loans? Good question. You see, the banks in America are private businesses and can have their own set of requirements. Actually, many financial institutions actively seek business of illegal immigrants and have allowed them to show their IDs from their native countries to do business with them. Many of them do not even ask if you are in the country legally or don't care if you are by simply taking that information into consideration (for instance, if they ask you about your immigration status and you honestly report your unlawful presence, they might still grant you a loan, they might just charge you a higher interest rate or ask for a collateral or insist on a co-signor who is a legal resident). For the same reason, the credit bureaus might start tracking your credit history either with your made up number or actually assign you a number by associating it with your address and name (the credit bureaus have zero interest in your legal status or SSN; they merely want to collect as much data as they can on an individual and will use a variety of identifying information -- remember they want to sell this data to other corporations and get paid for it).
So does it mean that it is legal to use a false SS# to apply for credit? Well, legally, no. Generally speaking, using a fraudulent document or even providing false information to anyone (including an individual; so providing false information to a landlord can come to bite you) can be used against you. In any case, in all business transactions like this, you will be signing a lot of papers in which you will agree to provide only truthful information and if you do not, they can drag you to court. In reality, though, if a bank knowingly takes your false information, starts a banking relationship with you, and you meet your end of the bargain (for example, you pay back all the money you owe according to the terms of the contract), most likely nothing will happen. The banks simply want their money back; they don't care that you provided false information. However, if something goes wrong, then, in addition to suing you for all kinds of things, they will also include a charge that you provided false information or lied or used fraudulent documents. That can get ugly. So should you still do it? Not if you want to follow the law.