What happens when you apply for job with false social security number?

Generally speaking it is not legal to apply for work or actually work if you are not authorized for employment by the Department of Homeland Security.  While each employer will have their own procedure for conducting a background check for each applicant or at least on those who they wish to actually hire (some will use eVerify to check your right to work while others may also use the services of background check firms that pull up complete information about you, criminal records, etc.), by law, once they are hired, the employer is required to notify the Federal and State governments about the identity of the employer.  Then, from each paycheck, the employer must make proper deductions and deposit them.  In addition, there are lots of other documents and forms that employers complete to let various government agencies know about their employees.  Depending on individual cases, illegal aliens can be identified at some stage, but many are able to work illegally for years without consequence.