The Boeing 777 carrying 239 flew far off course for unknown reasons after leaving Kuala Lumpur for Beijing on March 8, 2014.
An ongoing search of the southern Indian Ocean has found no trace of the plane, though a wing part from the aircraft washed ashore on Reunion Island last year.
A global aviation agreement sets a two-year deadline for lawsuits by next-of-kin over air accidents.
Lawyer Sangeet Kaur Deo, who is representing 10 families, said the lawsuits involve a passenger from Russia, one from China and the rest from Malaysia.
Sangeet said her clients were engaged in settlement negotiations with the airline earlier.
"Everyone waited till the last minute to give time to the airline to settle but nothing reasonable was forthcoming. So they have no choice but to take legal action given the time limitation," Sangeet said.
Lawyer Yeoh Cho Kheong, who is representing the families of two Ukrainian passengers, said his clients will continue negotiations with the airline despite the lawsuit. The two men, aged 44 at the time, were partners in a furniture business.
"If an out-of-court settlement can be achieved, I believe my clients will accept it," Yeoh added.
All the lawsuits are seeking unspecified sums for negligence and breach of contract, the lawyers said. Sangeet said she expects to file two more lawsuits on Monday, the last day for filing.
An international aviation agreement allows each next-of-kin of passengers on board a plane up to USD 175,000 in compensation, but a plaintiff filing a lawsuit can seek more.
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