The study, for Shelter Scotland and Shelter, also found that almost half (48%) of those surveyed were currently struggling or falling behind with their payments.
Around one in seven (15%) say they have not opened post if they thought it was a bill or late payment reminder, with one in 10 (11%) putting it straight in the bin without opening it.
Shelter Scotland warned that people were at risk of losing their homes and urged people struggling to pay their bills to get help and not adopt a “head in the sand” approach.
Graeme Brown, director of Shelter Scotland, said: “It’s a worrying sign of the times that so many people are starting the New Year worried about how they’ll pay their rent or mortgage in 2014.
“And it’s even more concerning that so many people are putting off dealing with or ignoring their situation. Unless they face up to the facts and get help soon, some of the households struggling now could face the very real prospect of losing their home this year.
“Despite recent discussion of an economic recovery, we know that a combination of high housing costs, wage freezes and rising food and energy bills has put many families and individuals on a knife-edge, pushing them to breaking point.
“We urge everyone struggling on that knife-edge to get help as soon as possible instead of ignoring the problem and making it much worse. We can help stop people’s lives spiralling out of control towards homelessness.”
The survey of 4,000 British adults, including 383 Scots, was carried out from 20-22 November 2013. It was carried out by YouGov for Shelter and Shelter Scotland.