In five areas of the country, more than half the adult population has savings below that level, reports The BBC.
Those areas are Northern Ireland, the West Midlands, Yorkshire and Humber, North East England and Wales.
The MAS said the findings were worrying and presented a particular challenge for low earners.
“These figures show the millions put at risk by the saving gaps in the UK,” said Nick Hill, money expert at the service.
“For some on low incomes, saving is a real challenge as they may simply lack the income needed to save at all.”
The research was carried out for MAS by the consumer data company CACI which has a database of 48m UK adults.
However, the research also showed that some people on low incomes do save money.
Roughly a quarter of adults with household incomes below £13,500 have more than £1,000 in savings.
And 40% of people in that income bracket manage to save something every month.
The MAS says saving small amounts on a regular basis is achievable for most people.
“Regular saving is key to building up that buffer against those life surprises,” said Mr Hill.
“If you earn enough to set even a little aside each month that’s great – a direct debit into a savings account might be an easy way to do this, even if you start small and increase the amount with time.”
Since April, basic-rate taxpayers have been allowed to earn up to £1,000 a year in a savings account, and pay no income tax.
The Personal Savings Allowance, as it is known, is worth £500 a year to higher-rate taxpayers.
