Overall, Cork will get the largest amount, with €1m assigned for 41 projects.
Galway is second with €942,000, followed by Donegal (€942,000), Mayo (€885,000), Kerry (€727,000) and Tipperary (€653,582).
The lowest payments are being made to the local authorities in Carlow and Louth, which will get €250,000 each.
The LIS is a scheme for carrying out improvements on private and non-publicly maintained roads.
Although not in the possession of the local authorities, they often lead to multiple homes or amenities such as lakes, rivers or beaches.
The Programme for Government, agreed by Fine Gael and independents, included a commitment to reinstate the fund, which was a victim of the recession.
It also formed part of the Government's Action Plan for Rural Development.
The €10m to be assigned this week comes on top of €17m announced last September.
"I cannot overstate how important local connectivity is in order to enable rural Ireland to function well," said Mr Ring (pictured right).
"The Action Plan for Rural Development prioritises improving rural infrastructure and connectivity, and continuing to support an LIS scheme is just one of the measures we are taking in that regard.
"Many lanes and boreens which link private homes and farms are not on the public road network and so are not maintained by the local authorities.
"The Local Improvement Scheme is therefore a very valuable support for rural families and businesses that use these roads on a daily basis."
Just last week, Tipperary TD Michael Lowry, who votes with the Government, told the Dáil that roads in some parts "have become virtually undriveable".
"They are not fit for purpose. They are hazardous. They are dangerous.
"In fact, they are obstacle courses," he said, adding that if Fine Gael "is serious about re-election it must address the lack of funding for rural roads because after the first week of the campaign half of the canvassers will be in bed because they will not want to face the hostility of the people living down rural roads".
Taoiseach Leo Varadkar rejected allegations that the Government is not doing enough, urging TDs to ask council officials how much of their resources have been deployed so far this year.
He said €417m was available for local and regional roads this year "and I understand only a fraction of that has been spent".
"I understand why that is, namely, because of the bad weather, but we want local authorities to get out there now in the good weather and with the longer evenings to spend the €417m improving and restoring our regional and local roads.
"In the meantime, the Government will work on additional funding following the storms."
Detailed information on the projects selected by local authorities for funding under the €10m scheme being announced today can be found on the Department of Rural Affairs and Community Development website.