there is an SSN work-authorization test one has to pass when applying for most jobs. No SSN or SSN not authorized for work (all temp|visitor visas) - gtfo.
Source - been there done that.
The whole immigrant debate is so political now, it's hard to find any trustful source, including the stats put out by USCIS , DOJ, etc.
So if you want to know how the SSN game works for illegal immigrants, find some media who wants to paint it in a positive light, even if they have to admit that millions of workers use fake SSNs: https://www.theatlantic.com/business/archive/2016/09/undocum...
