BBC report
Legal minds have their say
In a joint statement, 16 attorneys general, from states including California, New York and Pennsylvania, said they would "use all of the tools of our offices to fight this unconstitutional order" and, until it was struck down, would "work to ensure that as few people as possible suffer from the chaotic situation that it has created".
Late on Saturday, federal Judge Ann Donnelly, in New York, ruled against the removal from the US of people with approved refugee applications, valid visas, and "other individuals... legally authorised to enter the United States".
The emergency ruling also said there was a risk of "substantial and irreparable injury" to those affected.
She was ruling on a case brought by the American Civil Liberties Union (ACLU) on behalf of two Iraqi men with links to the US military who were detained at JFK Airport in New York.
Both have now been released. Another court hearing is set for February.
Elsewhere in the US:
In Boston, a judge decided two Iranian nationals, university professors, should be released from detention at Logan International Airport
An order issued in Virginia banned, for seven days, the deportation of green-card holders held at Dulles Airport and ordered the authorities to allow access to lawyers
A Seattle judge issued an emergency stay of removal from the US for two people
The Department of Homeland Security has said it will continue to enforce the measures.
Amateur hour at the White House? - analysis by Anthony Zurcher, BBC News, Washington DC
These rulings are only the opening salvo in what will likely be a protracted legal battle, as the Trump administration forges ahead with its plans.
The episode has made the White House look amateurish and ill-prepared, however, and Republicans in Congress are getting nervous.
During the presidential primaries, a majority of Republican voters backed Mr Trump's calls for a sweeping ban on Muslims entering the US. During the general election, he campaigned on a visa ban for certain "terrorist" countries - and won.
So it shouldn't come as much of a surprise if the president's core support holds firm after this weekend's events. The views in the American heartland, far removed from major international airports, sometimes differ greatly from the liberal bastions on the coast.
However, protracted airport detention of children and the elderly is "bad optics" - and could make it harder for the White House to get public support for future immigration action.