The part-taxpayer-owned bank and its sister, NatWest, took two of the bottom five places along with HSBC and Lloyds Bank.
Both RBS and Natwest have been beset with a string of high-profile technical glitches in recent years, including a cyber-attack that prevented account holders accessing online banking and a fault in the run-up to Christmas last year that meant customers' credit and debit cards stopped working for several hours.
The poll results come a day after the Edinburgh-based bank was fined £14.5 million for "serious failings" in its mortgage business.
The results were compiled by MoneySavingExpert.com for its twice-yearly survey.
First Direct has retained the top spot in the customer satisfaction poll of current account providers, while Barclays languishes in bottom place for the first time.
Santander, Nationwide, the Co-operative Bank and its internet bank brand Smile rounded out the top five.
Lloyds, RBS, HSBC, NatWest and Barclays took the bottom five places. Fewer than half of consumers rated the current account service provided by the five as great.
First Direct, which is an internet and telephone banking brand belonging to HSBC, has consistently topped MoneySavingExpert's consumer survey since it started in 2008. More than nine in 10 (92 per cent) of consumers rated First Direct's current account service as great.
A Barclays spokesman said the findings did not tally with its own research into how happy its customers were.
Santander, which spent four years at the bottom of the survey, has shot up the rankings into second place, with 72 per cent of customers saying it gives a great service. Despite the Co-op's recent woes from its financial turmoil and the drugs scandal involving former chairman Paul Flowers, its customers ranked the service they receive from the bank highly.
Smile came third in the survey, with 72 per cent of customers describing the service as great, while the Co-op came fifth, after 70 per cent of its customers gave it the thumbs up.
The Co-op recently launched a fightback for the customers it shed in the aftermath of its scandals, including conducting a widespread customer survey around its ethical policy, which it plans to relaunch soon.
Meanwhile, TSB now sits higher than Lloyds in terms of consumer satisfaction, according to the results.
Lloyds TSB split into two separate banking brands last autumn, but on the first day of TSB's big re-launch, TSB and Lloyds were hit by an embarrassing glitch that left customers unable to log on to internet banking. Both banks have revamped their current account deals since splitting.
Some 7,875 votes were cast in total for the survey, and banks with fewer than 100 votes were not included in the results.
New rules came into force last year that have cut the maximum length of time it takes to switch a current account to seven working days.
Guy Anker, managing editor at MoneySavingExpert.com, said: "With current accounts, customer service really counts. Whether you bank online or in-branch, this is the one financial product you do genuinely have a real day-to-day interaction with.
"Some banks are delivering the goods with both innovative accounts as well as good customer service. Just look at Santander; in 2012 it was bottom of the table but it has stepped up."