Ninety-three of the retailer's 125 stores will close, with the rest remaining open while buyers are sought.
If administrator KPMG cannot find buyers, the remaining stores will also shut. The chain employed 1,658 people.
The chain was sold to Greybull Capital last year by supermarket Morrisons, which has said it would offer jobs to ex-My Local staff.
It has struggled to compete in the cut-throat grocery sector, which is caught in a fierce price war.
My Local, formally known as Morrisons' M Local, was sold to Greybull for Â£25m in a deal fronted by retail veteran Mike Greene, who appeared on Channel 4 show The Secret Millionaire.
Earlier this month My Local said that administrators had been put on notice to take over running the business.
Last week Morrisons offered a lifeline to My Local staff, saying it wanted to offer them jobs in its supermarkets if there were store closures.
Mr Greene said: "The last nine months has been one of the toughest retail trading periods that I have ever experienced and My Local has faced intense competition.
"The management team has been unable to return the business to profitability. For that I am truly sorry."
Mark Orton, partner at KPMG and joint administrator, said: "Management have faced tough trading conditions and despite their best efforts to improve performance, My Local was ultimately unable to return to viability.
"Although the majority of stores have now closed, we are in active discussions with a number of interested parties in relation to both the remaining 32 premises that are trading and a small number of those that have closed," he said.
The failure of My Local comes at a difficult time for the retail sector, following on from the failures of BHS and Austin Reed.
On Tuesday, the owners of fashion firm Store Twenty One said they would push for a restructure that could result in store closures and job losses.
Convenience store chain My Local has gone into administration, with more than 1,000 jobs likely to be lost.