Legal filings hint at the broad constitutional and statutory arguments to come, but judges have not yet ruled squarely on the merits one way or the other.
The challenges began a week ago when individuals with valid visas, among others, found themselves in legal limbo. They were in transit when Trump's order went into effect on the evening of January 27 and they were not immediately allowed into the country.
Now several states are also bringing claims, including Washington, Minnesota, Massachusetts and Hawaii among others.
The claim brought by attorneys general in Washington and Minnesota -- the one that led federal Judge James Robart in Seattle to put a temporary nationwide halt to the travel ban Friday night -- is the furthest along so far, but the legal landscape changes quickly.
Trump's order bars citizens of seven Muslim-majority countries -- Iraq, Syria, Iran, Libya, Somalia, Sudan and Yemen -- from entering the US for 90 days, all refugees for 120 days and indefinitely halts refugees from Syria.
In order to bring suit, states must prove that they have been harmed.
In court papers, attorneys general for Washington and Minnesota argue they have the legal right to be in court.
"Washington has declared that practices that discriminate against any of its inhabitants because of race, creed, color or national origin are matters of public concern that threaten the rights and proper privileges of the State and harm the public welfare," they wrote. They also claim the executive order would hurt the state's economy, pointing to Microsoft, which is headquartered in the state.
"Microsoft's US workforce is heavily dependent on immigrants and guest workers," they write. "At least 76 employees at Microsoft are citizens of Iran, Iraq, Syria, Somalia, Sudan, Libya or Yemen and hold temporary work visas."
Minnesota stresses that the order would impact the Mayo Clinic that is based there.
"The Mayo Clinic, an internationally known health care institution that treats patients who travel to it for health care services from around the world, has publicly stated that approximately 80 staff, physicians and scholars have ties to the affected countries."
The government says the states don't have the right to be in court, and on the threshold issue, government lawyers say the states are attempting to manufacture standing.
In court papers, lawyers for the government argue that "lost tax revenues, or reputational injury to its universities -- are not concrete, particularized harms."
And the lawyers say that the states can't rely on any injuries to lawful permanent residents because the White House has clarified that the 90-day suspension of entry does not apply to those individuals.
"In effect, the states are arguing that, unlike individuals temporarily detained at the airport, they will suffer a longer-term economic injury from the executive order -- one that therefore allows them to challenge it," said Steve Vladeck, CNN legal analyst and professor of law at the University of Texas School of Law.
"This is a controversial argument, but, more than a little ironically, it's based on similar reasoning to what a number of more conservative states, led by Texas, successfully used in the lower courts to challenge President Obama's deferred action immigration program," he said.
Constitutional and statutory arguments
Moving to the merits, the states make a variety of statutory and constitutional arguments.
Washington and Minnesota argue that the executive orders "target individuals for discriminatory treatment based on their country of origin and/or religion, without lawful justification" and they say the order was "motivated by animus and a desire to harm a particular group."
They also say that the order violates the Establishment Clause that prohibits the federal government from preferring one religion over another. They say parts of the order, coupled with some statements made by the President in the press, "are intended to disfavor Islam and favor Christianity."
And they make a due process claim arguing that the federal government cannot deprive individuals based on their liberty interests without due process of the law.
Also in play is the Immigration and Nationality Act of 1965, which they argue "prohibits discrimination in the issuance of immigrant visas on the basis of race, nationality, place of birth, or place of residence."
Hawaii Attorney General Doug Chin echoed these arguments in a different case.
He argues the executive order "is subjecting a portion of Hawaii's population to discrimination and second-class treatment, and denying them their fundamental right to travel overseas. Moreover, the executive order is eroding Hawaii's sovereign interests in maintaining the separation between church and state and in welcoming persons from all nations around the world into the fabric of its society."
"There are so many different legal objections to the executive order -- and so many different changes in policy in the order -- that it's hard to know which ones will stick," said Vladeck.
"The whole order is potentially vulnerable to the religious discrimination claims, whereas the due process objections are more to how individual sections of the order restrict the liberty of different classes of immigrants. But there's almost no way the entire order survives these lawsuits unscathed," he said.
In the hours after Robart issued the nationwide order blocking the ban, Trump was quick to respond.
"The opinion of this so-called judge, which essentially takes law-enforcement away from our country, is ridiculous and will be overturned!" Trump tweeted Saturday morning.
In court papers, his lawyers have so far been more measured.
On the merits, they note that Congress "expressly has delegated to the President the broad power to suspend entry 'of any class of aliens into the United States.' "
"The order was well within the President's authority under Congress' delegation, particularly in an area, like immigration, in which the admission to the United States of foreign aliens is subject to plenary control by the political branches," they said.
In issuing his temporary restraining order, Robart, the US district judge for the western district of Washington, acknowledged that issuing it is "an extraordinary remedy" but he held that the executive order "adversely affects the states' residents in areas of employment, education, business, family relations, and freedom of travel."
He said the harms are "significant and ongoing," and concluded that the order against the government was necessary until the court could hear arguments for further relief.
His order came after a different judge in Massachusetts declined to renew a temporary restraining order previously issued. Judge Nathaniel M. Gorton of the U.S. District Court for the District of Massachusetts said one of the factors he had to consider was whether the plaintiffs were likely to succeed on the merits.
"Because plaintiffs have not demonstrated that they are likely to succeed on the merits of their claims, an extension of the restraining order at the present time is not warranted," he said.