While black box recorders are yet to yield the cause, Singapore and Australia became the latest nations to suspend Boeing 737 MAX aircraft on Tuesday.
Sunday's disaster — following another fatal crash of a 737 MAX jet in Indonesia five months ago — has caused alarm in the international aviation industry and wiped billions of dollars off the market value of the world's biggest planemaker.
Safety experts say it is too early to speculate on what caused Sunday's crash or whether the two recent accidents are linked. Most accidents are caused by a unique chain of events combining human and technical factors.
The 157 victims came from more than 30 different nations, and included nearly two dozen UN staff.
Given the problems identifying them at the charred disaster site, Ethiopia Airlines said it would take at least five days to start handing remains to families.
"We are Muslim and have to bury our deceased immediately," Noordin Mohamed, a 27-year-old Kenyan businessman whose brother and mother died, told Reuters.
"Losing a brother and mother in the same day and not having their bodies to bury is very painful," he said in the Kenyan capital Nairobi where the plane had been due.
Ethiopian Airlines flight 320 came down in a field soon after takeoff from Addis Ababa on Sunday, creating a fireball in a crater. It may be weeks or months before all the victims are identified.
The United States has said it is safe to fly the planes, and Boeing has said there is no need to issue new guidance to operators of the aircraft based on the information it has so far.
But Singapore and Australia's aviation authorities — following China, Indonesia and others — said temporary suspension of Boeing 737 MAX aircraft in and out of their airports was necessary during a safety review.
Anxiety was also evident among travelers, who rushed to find out from social media whether they were booked to fly on 737 MAX planes — the same model involved in the Lion Air crash off Indonesia that killed 189 people in October.
Black box recorders were found from the Ethiopian crash site on Monday, but it was unclear where they would be looked at.
So long as the recordings are undamaged, the cause of the crash could be identified quickly, although it typically takes a year to fully complete an investigation.
Nearly 40 percent of the in-service fleet of 371 Boeing 737 MAX jets globally is grounded, according to industry publication Flightglobal. That includes 97 jets in biggest market China.
Boeing shares fell 5 percent on Monday.
The US Federal Aviation Administration (FAA) issued a "continued airworthiness notification" for the 737 MAX late on Monday to assure operators, and detailed a series of design changes mandated by Boeing in response to the Indonesia crash.
Boeing issued a statement as well, saying it had been working with the FAA in the aftermath of the Lion Air crash to develop enhancements to flight control software that will be deployed across the 737 MAX fleet in coming weeks.
The MAX 8 has new software that automatically pushes the plane's nose down if a stall is detected.
The new MAX 8 variant of the 737, the world's best-selling modern passenger aircraft, has bigger engines designed to use less fuel. It entered service in 2017.
With another 4,661 on order, 737 Max 8s could become the workhorses for airlines around the globe for decades.
Ethiopian Airlines, which has four other 737 MAX 8 jets, has grounded them as a precaution. A prize-winning author, a soccer official and a team of humanitarian workers were among those who perished on its flight.
Gol in Brazil temporarily suspended MAX 8 flights, as did Argentina's state airline Aerolineas Argentinas and Mexico's Aeromexico.
South Korean budget carrier Eastar Jet said it will temporarily ground its two 737 MAX 8s from Wednesday to cooperate with the government's emergency safety inspections.
India's regulator ordered additional maintenance checks on 737 MAX 8 aircraft operating in the country and said a review found "no significant concern."
Vietnam state media reported the aviation regulator would not issue licenses to local airlines to operate the 737 MAX until the cause of the Ethiopian crash is determined.
Still, major airlines from North America to the Middle East kept flying the 737 MAX, though Canadian Transport Minister Marc Garneau said he would not hesitate to take action once the cause of the crash is known.
Southwest Airlines, which operates the largest fleet of 737 MAX 8s, said it remained confident in the safety of all its Boeing planes even as it fielded queries from customers.
Former FAA accident investigator Mike Daniel said the decision by regulators to ground the planes was premature. "To me it's almost surreal how quickly some of the regulators are just grounding the aircraft without any factual information yet as a result of the investigation," he told Reuters.
By Duncan Miriri and Jamie Freed/Reuters
Additional reporting by Aradhana Aravindan in Singapore; Katharine Houreld in Nairobi; Eric Johnson in Seattle; James Pearson in Hanoi; Alexander Cornwell in Dubai; Heekyong Yang in Seoul; Tracy Rucinski in Chicago; Tim Hepher in Paris; Writing by Sayantani Ghosh and Andrew Cawthorne; Editing by Simon Cameron-Moore, Georgina Prodhan.