However, one in 10 don't think they need a cash cushion, leaving them and their families dangerously exposed if their financial situation changes.
The latest quarterly savings monitor from Lloyds Bank shows people's willingness to save has increased over the past 12 months, with those stating they save either regularly or whenever they can rising by 2 per cent to 67 per cent. Around the same proportion expect to put aside a similar or larger amount over the coming year.
Andy Bickers, Lloyds Bank's savings director, said: "It is encouraging to see a more positive outlook and shift towards savings in the last year."
Figures from NS&I support the bank's findings.
The Treasury-backed savings provider says people are now typically putting away £101, or 8 per cent of their income, each month, taking the average above £100 for the first time in a year.
But a survey by account provider Think Money highlights the fact that many are still without a rainy day fund.
It reveals that while many of those not saving say they can't afford to, a tenth of the adult population are choosing not to because they don't see the need.
Think Money spokesman Ian Williams said: "It is worrying that as many as one in 10 people don't realise the importance of putting money aside to build up a savings pot.
"While there will always be some people who really can't afford to save, for most of us, the decision to save is more about deciding to forgo some spending.
"Rather than thinking of saving as something we could do if we have money left over, it is key that people see it as a priority, just like paying their bills, rent or mortgage."
Someone without savings would have to overdraw, borrow on plastic or take out a loan to meet an unexpected expense such as a major domestic or car repair, leaving them to make repayments for months or even years to come.
And if the worst happened and their household income was slashed by job loss, illness or death, with only meagre state benefits coming in and insufficient funds to cover monthly essentials, they could soon find themselves homeless and drowning in debt.
To prevent this kind of disaster, everyone should aim to have the equivalent of at least three months' earnings ready in an easily accessible account.
Despite interest rates being at historic lows, there are wide variations between accounts, making it well worth shopping around for the best possible return.
Several current accounts pay competitive interest. Nationwide's FlexDirect offers 5 per cent (4 per cent after tax) for 12 months on balances up to £2,500 for those putting in at least £1,000 a month, and TSB Classic Plus pays 5 per cent on balances up to £2,000 without a time limit to those depositing £500 each month.
Meanwhile, Santander's 123 account, which has a £2 monthly fee and a minimum funding requirement of £500, pays up to 3 per cent (2.4 per cent after tax) on balances between £3,000 and £20,000.
For those with larger amounts to put away or who prefer to keep their savings separate from their day-to-day cash, the first port of call should be an instant access cash Individual Savings Account.
Unlike ordinary accounts, where - depending on the account holder's tax bracket - at least 20 per cent of the interest is taken off, interest earned on Isas is tax free, maximising the return.
Everyone over the age of 16 can open a new one each tax year.
The current maximum annual deposit is £5,940 but this will rise to £15,000 from July 1.
Several providers, including Virgin Money and NS&I, have instant access Isas paying long-term rates of around 1.5 per cent. Halifax has one offering 1.55 per cent per cent but this drops to 0.25 per cent after 12 months, so anyone choosing it should be prepared to move their money at this point.
Those who have already used their annual Isa allowance can earn similar rates after tax with two to six-month notice accounts from a range of smaller institutions.
These include Shawbrook Bank, Islamic Bank of Britain and Raphaels Bank.
And for anyone with enough savings to leave a proportion untouched for a year or more, the same providers have bonds paying pre-tax rates of around 2 per cent. Credit unions such as Scotwest, Glasgow, Capital and others, which are open to anyone living or working in their catchment area, also have instant access and notice accounts paying from 1.5 per cent to 2 per cent pre-tax.
Mr Bickers at Lloyds said: "With one in three saying that they're still not saving anything, there is still some way to go.
"Even if people just put away a small amount each month, this can be increased as circumstances improve."