The number of complaints about current accounts shot up by nearly one third in the first half of 2015, according to figures from the City regulator.
The data also shows that pensions-related complaints jumped by around one fifth as the new retirement freedoms were brought into force.
More than half a million complaints were made about current accounts between January and June, making them the second-most complained about financial product after PPI (payment protection insurance), according to the Financial Conduct Authority (FCA).
Some 506,326 complaints into current accounts were recorded, marking a 31.2% increase compared with the previous six months.
Meanwhile, 73,055 pensions-related complaints were made in the first half of 2015, a figure which is up by 19.7% on the second half of 2014.
Changes were brought into force in April to give people aged 55 and over more flexibility over how they use their pension pot, although there have been reports of some people struggling to access their money in the way they want to.
Overall, financial services firms received 2.1 million new complaints between January and June 2015. This was a 2.1% decrease compared with the previous six months, driven by a fall in PPI-related complaints.
FCA director of strategy and competition Christopher Woolard said: "While the ongoing fall in PPI complaints is welcome, this is the second half-year running that we have seen complaints about banking products rise.
"It is clear that firms need to look at the causes for this rise and where necessary take action to address the causes of the trend.
"Ensuring good consumer outcomes should be at the heart of firms' activities and we want to see complaints fall in the future as firms seek to ensure that consumers get the right products and services."
The number of complaints about PPI fell below one million for the first time since 2012, according to the half-year figures. Some 883,043 such complaints were recorded, down 16.6% compared with the previous six months.
The FCA's figures also show that Barclays received the most complaints about banking and credit cards in the first half of 2015, with 140,574 issues recorded, followed by NatWest which received 95,994 complaints about these products.
The Financial Ombudsman Service, which resolves disputes between consumers and financial firms, recently said it is receiving around 1,000 new complaints about packaged current accounts every week.
These accounts generally charge consumers between £5 and £25 a month in return for a bundle of perks, but some people have told the ombudsman they found they were not eligible for the perks. In other cases, consumers have claimed they were signed up to a packaged account they never wanted.
Which? executive director Richard Lloyd said banks must step up efforts to make sure they deal with issues quickly and fairly.
He said: "The decrease in the overall number of complaints is driven by a drop in PPI claims, but overall the figures are still far too high and it's particularly worrying to see current account complaints continuing to rise."
Yvonne Braun, director of long-term savings policy at the Association of British Insurers (ABI), said: "Pension providers have dealt with an unprecedented number of enquiries since the introduction of the pension freedoms in April - up 80% in the first month alone.
"Almost £2.5 billion was paid to customers in the first three months, showing huge numbers of people have accessed their pensions successfully.
"Where customers have been frustrated by requirements to get formal financial advice or have struggled to understand their options, the ABI has proposed solutions to policymakers."
According to the FCA's figures, the top five most complained about products and services in the first half of 2015 were:
1. Payment protection insurance (883,043 complaints - down 16.6% compared to the previous six months).
2. Current accounts (506,326 complaints - up 31.2% compared to the previous six months).
3. Other general insurance (304,919 complaints - down 4.3% compared to the previous six months).
4. Credit cards (121,615 complaints - up 11.1% compared to the previous six months).
5. Savings, including cash Isas (78,981 complaints - down 6.4% compared to the previous six months).
The FCA said the top five most complained about firms, in terms of the number of complaints received in the first half of 2015, were:
1. Barclays Bank (283,221 complaints).
2. Lloyds Bank (232,971 complaints).
3. Bank of Scotland (190,121 complaints).
4. NatWest Bank (144,741 complaints).
5. HSBC (140,244 complaints).