In New Haven, Conn., on Monday, three people were arrested for fighting in line. One of the three was taken to hospital, media reports said.
"There seem to be two groups, rival groups, mainly of Chinese people who are coming in from New York on a daily basis and trying to buy as many of the phones as the store will allow to sell to any one person," New Haven police spokesman Officer David Hartman told NBC News.
Buyers are able to resell the phones, which go for about $1,000 each unlocked with no phone-company contract, for as much as $3,000 to $4,000 apiece, police said.
"These rival groups have been challenging each other over their places in line," Hartman told NBC. "At times, these challenges have resulted in violence."
In Paris, a man was restrained by French police after a scuffle broke out at an Apple Store, the Daily Mirror reported at its website.
And in British Columbia, police were called in to control 600 people lined up for the launch of Apple's new iPhone at Metrotown Mall in Burnaby, after the crowd surged toward the gates, overwhelming 10 security guards.
Apple says it has set a sales record with the release of its latest iPhone models.
It says it has sold more than 10 million iPhone 6 and 6 Plus models in the three days since they went on sale. That eclipses the record set a year ago, when Apple moved nine million of the iPhone 5C and 5S models.
The newest iPhone is available in the U.S., Australia, Canada, France, Germany, Hong Kong, Japan, Puerto Rico, Singapore and the U.K.
The phone will go on sale in 20 more countries on Friday — and others by the end of the year.