The areas include two-thirds of authorities in London, more than half in south-east England and nearly a half in eastern England.
Most of the rest of the country has yet to reach record levels, however, with only a handful of areas in the north and west seeing rates at an all-time high.
The figures, which have been compiled by the PA news agency, show that:
• 21 of the 32 local authorities in London now have record Covid-19 case rates, with the capital accounting for the top 10 highest rates in the UK and 20 of the top 25.
• Four London areas have rates above 2,000 cases per 100,000 people: Lambeth (2,461.4), Wandsworth (2361.9), Hackney and City of London (2,096.8) and Southwark (2,064.0)
• In south-east England, 37 of the 64 local authorities are now seeing record rates, led by Elmbridge (1,384.7), Reigate and Banstead (1,317.3) and Epsom and Ewell (1,271.6), all of which are in Surrey.
• 21 of the 45 local authorities in eastern England have record case rates, including St Albans (1,311.3) and Cambridge (1,177.0).
These are not the highest rates in the region, however – Brentwood (1,460.3) and Thurrock (1,342.2) in Essex are higher, though this is slightly below the record for both areas, which was set during the second wave of the virus last winter.
• 12 of the 40 areas in the East Midlands are at a record high, led by South Northamptonshire (970.8), Broxtowe in Nottinghamshire (917.8) and Charnwood in Leicestershire (916.6)
All figures are for the seven days to 16 December, as data for more recent days is still incomplete.
In total, 102 of the 377 local authority areas in the UK (27%) are now recording their highest Covid-19 case rates since mass testing was rolled out across the country in May and June 2020.
Figures for case rates in the early months of the pandemic are not directly comparable, as only a small number of people were being tested, mostly in hospitals and care homes.
Of the 102 areas, only 11 are outside the south and east: six in north-west England (Bury, Cheshire West and Chester, Manchester, Salford, Stockport and Trafford); three in Scotland (East Lothian, Edinburgh and West Lothian); one in Northern Ireland (Ards and North Down); and one in the West Midlands (Newcastle-under-Lyme).
The contrast between the south and east and the north and west reflects the way the Omicron variant of Covid-19 has spread in recent weeks – in particular, how London was the first area of the UK where Omicron became the dominant variant of the virus.
Although nine in 10 local authorities in the UK are recording a week-on-week rise in rates, most areas in the north and west of the country have yet to hit levels seen during the second wave of the virus.
But this could change in the days and weeks ahead, once Omicron has become the dominant variant in all parts of the country.