When going through the US immigration system, as part of the green card application you need to specify a sponsor who will be responsible for you financially if you hit hardship. As an immigrant, you do not get welfare in the US.
That's not an answer to the point above. Having people who live in our society but are ineligible for our social benefits is harmful and destabilizing to us.
Also note that people who are in the US on nonimmigrant visas still have to pay SSDI taxes, even though they are ineligible for the benefits. (Though they get to count them towards the time needed for sponsorship if they become immigrants.)
Depending on the state non-immigrants may be eligible if they've paid into the system as generally the state benefits are for state residents, a status that even as a non-immigrant you qualify for.
However most visa categories require you to leave the US within ten days if you no longer meet the visa requirements. So it's difficult to claim said benefits without messing up your future immigration chances.
http://www.nydailynews.com/new-york/obligations-affidavit-su... - This details at least one court case of USCIS enforcing an Affidavit of Support, and was literally the first Google result on which I clicked. They do mention that it's rare, but it happens.
No, that's an immigrant enforcing the affidavit of support as a third party beneficiary against the sponsor, I've heard of cases like that, though they are relatively rare.
What I was talking about was where a government agency sues to recoup funds when a beneficiary uses in a means tested program. That I've never seen and google doesn't seem to come up with any examples.
Source: US immigrant, did the paperwork.