I'm pretty sure foreign residents get an Individual Tax ID Number (TIN). [1] Not sure why @swalberg doesn't have one...
[1] https://www.irs.gov/Individuals/International-Taxpayers/Taxp...
I applied for a SSN when I got here but it took 4 weeks to process. I now have an SSN card with a statement on it saying that I'm only allowed to work with DHS authorization.
Part of the hassle is because I signed up for services with no SSN but web sites all assume you have an SSN. The other problem is that credit histories don't cross the border. Even with an SSN, I've got no public record or credit history.
What I find funny (or sad) is that when I call to update my records with my SSN (largely so I can build a credit history) people refuse to take the number over the phone, and will not take an email scan. They only want snail mail or fax.