According to the US immigration laws, permanent residents must keep their Green Cards with them at all times. They need to carry with them their original permanent residents cards. But they can make copies of their cards and store those copies in a safe place. This will help them to replace their Green Cards if they lose them.
This also applies to the other immigration documents and the permanent residents must keep their passports in a safe place. Only while traveling abroad they need to carry their original passports with them.
While traveling abroad, it is wise to make copies of all the immigration and travel documents and carry copies of those documents. This will help if they happen to lose their original documents while traveling abroad. They can also leave copies with their family members or relatives in the United States. These copies will help them to get their original documents replaced if the originals are lost or stolen.
Likewise, non-immigrants also must make copies of their passports and their non-immigrant visas. They can use those copies when they are encountered by immigration officers, to establish that they are not undocumented immigrants.
Green Card holders and the naturalized US citizens must both retain copies of their immigration papers. Just like their marriage certificates, birth certificates, tax records and insurance policies, they need to keep these documents in a safe place.
Many carry with them their original Social Security cards. But that is not mandatory and it would be sufficient if they just note their Social Security numbers on a piece of paper and keep it with them. They are likely to become victims of identity theft, if they lose their Social Security cards. Likewise, they must not provide these numbers to anyone who calls them and requires them to provide these numbers.
They need to remember that they must file applications and wait for a long time to get their lost documents replaced. It is possible to get a lost Green Card or a lost employment authorization document replaced. But the applicants will be put on a waiting list and they cannot get their documents immediately after filing applications. In order to avoid such difficulties, people must see to it that they do not lose their documents and they need to keep them in a safe place.