* Graphic: World FX rates in 2020 http://tmsnrt.rs/2egbfVh

* Graphic: Trade-weighted sterling since Brexit vote http://tmsnrt.rs/2hwV9Hv

LONDON, Oct 26 (Reuters) - Sterling stabilised against the dollar on Monday as markets turned cautious over new lockdown restrictions in Europe and record-high daily virus cases in the United States, with riskier currencies losing out.

U.S. COVID-19 cases hit record highs over the weekend, with hospitalisations at a two-month high. In Europe, Spain announced a new state of emergency and Italy imposed new lockdown measures to curb a second wave of infections.

Equities fell and the safe-haven dollar rose as a wave of risk-aversion swept markets. The euro was down around 0.5% against the dollar.

At 0935 GMT, the pound was flat against the dollar at $1.3043. It was up around 0.4% against the euro, at 90.54 pence.

Brexit negotiations remain in focus for pound traders, as the United Kingdom and the European Union have just over two months left to agree on their future trading relationship, ahead of a Dec. 31 deadline.

The EU's chief negotiator, Michel Barnier, has extended his stay in London for intense Brexit trade talks until Oct. 28 - news which had limited impact on sterling.

"News that UK-EU trade talks have been extended until 28 October ushers in a little more optimism that some kind of free trade deal can be secured," wrote ING FX strategists in a note.

"Given the UK's economic underperformance, however, prompting more Bank of England quantitative easing in November and possibly negative rates next year, we doubt a deal will trigger a major re-assessment of GBP prospects," they added.

But there are some hopes that a Brexit deal can be reached, with Ireland's deputy Prime Minister and Northern Ireland minister Brandon Lewis expressing optimism on Sunday.

Goldman Sachs FX strategists said that Brexit developments were looking positive.

"Despite sterling's underperformance on Friday, we continue to see a path for a sharply higher GBP/USD cross into year-end," they wrote in a note.

The EU's markets watchdog said on Monday banks and asset managers based in the bloc should execute most of their share trades inside the EU after full Brexit from January, in a step set to fragment cross-border markets.

Elsewhere, news that a vaccine being developed by British drugmaker AstraZeneca and the University of Oxford produced an immune response in elderly and young people failed to bolster market sentiment.

Britain's health minister said his central expectation was for a vaccine to be rolled out for the first half of 2021. (Reporting by Elizabeth Howcroft; editing by Emelia Sithole-Matarise)