What's the legality of Muslim Registry suggested by some of Trump's administration and his supporters? Does this violate freedom of religious expression or other federal discrimination laws?
Any law requiring registration on the basis of religion would impose a burden based on religion. Such a law which makes a suspect classification conflicts with the First Amendment, so it would be subject to strict scrutiny. One of the requirements of such a law is that it be justified by a compelling government issue – national security is the clearest such interest. A second is narrow tailoring, i.e. that it be written to just do that one thing that justifies its existence, and the third is that it be the least restrictive means (these are technically considered to be distinct, but that's judge talk and not a common-sense obvious distinction). Since there is no proposal to require registration of Muslims, it's difficult to be much more specific about exactly why such a law would be ruled unconstitutional. Clearly there are less restrictive means of e.g. preventing ISIS members from entering the country than requiring everybody to register their religion or lack thereof with the government. While we don't know exactly what the constitutionally protected "right to privacy" relied on in Roe v. Wade, it is beyond doubt that personal religious beliefs are not the government's concern, so even asking the question would be unconstitutional.