(Updates after UK data, adds new quote)
By Jemima Kelly
LONDON, Sept 30 (Reuters) - Sterling stayed below $1.30 on Friday and was on track for a fifth consecutive quarter of losses - the currency's worst run since 1984.
The pound plunged to a 31-year low after Britain voted to leave the European Union, falling as low as $1.28 early in the third quarter, having already weakened in the run-up to the June referendum on worries about its outcome.
Sterling is now trading more than 40 U.S. cents - or 25 percent - lower than the six-year highs it reached in mid-2014, as expectations for the Bank of England to hike interest rates have dried up.
Last month the BoE cut its key interest rate to another record low and relaunched an asset-purchase programme in an effort to cushion the blow dealt by the vote for Brexit, and some expect it to ease policy again before the end of the year.
Data showing Britain's giant services sector grew much more strongly than expected in July, in the clearest sign to date that the economy did not slow sharply after the shock of the referendum, lifted sterling a little but could not push it above $1.30. It traded up 0.1 percent at $1.2982 by 0845 GMT.
"Certainly some of the data has been better but I think that's a longer-term concern about the economic dislocation caused by Brexit," said Societe Generale currency strategist Alvin Tan.
"This has not been lifted, so it's going to difficult to see a strong bounce in sterling until we get much more clarity on the direction of what the post-Brexit economic regime will be."
A survey published earlier on Friday showed British consumer morale rocketed back to pre-Brexit levels in September, confounding expectations that the vote to leave the EU would wreak more lasting damage on Britons' willingness to spend.
"The good news is that from the perspective of UK consumers, the Brexit shock has been fleeting," wrote Bank of Tokyo-Mitsubishi UFJ currency strategist Derek Halpenny.
Against the euro, which was down against most currencies on the back of worries over the health of Deutsche Bank, sterling climbed a third of a percent to 85.26 pence, less than a penny away from a six-week low of 87.16 pence hit on Monday.
Next week, manufacturing and production data should provide indicators of the health of the British economy. (Editing by Dominic Evans)