Several US, Malaysian and Australian law firms told AFP they had begin filing suit on behalf of dozens of relatives of the 239 people on board the flight, seeking undisclosed damages.
Under international agreements, families have two years to sue over air accidents.
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But some also plan to sue Malaysia's civil aviation authorities and military for losing track of MH370, attorneys said, and at least one will target aircraft manufacturer Boeing.
If successful, total payouts could reach into the "hundreds of millions" of dollars, said Joseph Wheeler, an Australian attorney who is seeking an out-of-court settlement for four Malaysian next-of-kin before the deadline.
"It's a full Boeing 777, so the claims will be substantial," said Wheeler.
The overnight flight from Kuala Lumpur to Beijing vanished soon after take-off on March 8, 2014.
Investigators believe it diverted and crashed in the southern Indian Ocean, but neither a crash site nor cause have been found.
Under international agreements, families automatically get compensation totalling around $160,000 per passenger.
But they can sue for more, and Malaysia Airlines must prove it was not at fault, a tough task given the lack of clues.
Among major cases, 43 people, nearly all Chinese, sued in New York in early February, according to a copy of their case. MH370 carried 152 Chinese nationals.
A Miami-based firm is moving ahead with various lawsuits involving nearly 200 next of kin from several countries, said their attorney Roy Altman.
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