The only rights that permanent residents don't have are: a) the right to vote, b) the right to have their permanent residency not revoked for cause (e.g., committing a serious crime). Permanent residents get to become citizens after some time.
Non-permanent residents have fewer rights, and which rights they don't have is generally a matter of law, not Constitution, except that they have all the rights of Due Process guaranteed by the Constitution (i.e., by far the most important ones for them to have), at least as to criminal prosecutions, and often as to civil suits. A typical example would be the right to keep and bear arms, which non-residents generally do not have in most States. I suppose even First Amendment rights might be curtailed for non-residents, though I suspect at most only freedom of association would be, and only indirectly.
But most of the Bill of Rights (the 4th through the 8th Amendments) deals in Due Process, and these rights are available to all persons in the U.S., except perhaps as to immigrant law itself (e.g., a non-resident alien may have much less recourse to fight deportation than they and anyone else might have to fight a conviction).
From Wikipedia, some other examples are the right to be considered for federal employment (most Federal Government jobs being closed to non-citizens), access to US consular officials if arrested abroad, and exemption from certain real estate taxes[1].
[1]: https://en.wikipedia.org/wiki/Foreign_Investment_in_Real_Pro...
The Constitution distinguishes natural rights of all persons vs rights and privileges of "the people" being citizens and non-citizen legal residents.
AFAICT, the Federal courts have not generally adopted this view (though it is popular in some circles, particularly the US Department of Justice); ruling less specifically that the term “the people” instead “refers to a class of persons who are part of a national community or who have otherwise developed sufficient connection with this country to be considered part of that community.” United States v. Verdugo-Urquidez, 494 U.S. 259, 265 (1990).
Would you have more reading on this? It's something I've wondered about but never really come across a comprehensive answer to. What rights do noncitizens not have that citizens do?