The gradual improvement in the country’s transport came as snowploughs fought to clear tracks and major roads, with the A68 link into England at Carter Bar in the Borders being restored.
However, police warned motorists not to ignore road closed signs after several drivers attempted to struggle through snowbound roads only to get stuck and call for help.
Sergeant Mike Gallie said: “It is important motorists listen to these warning messages. Roads are closed for good reason.”
The first cross-border passenger train since Wednesday left Edinburgh at lunchtime, with a reduced timetable planned by Virgin Trains East Coast today.
The West Coast main line south from Glasgow remains shut as far as Carlisle, with no replacement buses. Last night there were no plans to operate services on the line today.
ScotRail expects to open all its routes today except possibly Inverness-Wick/Thurso.
A spokeswoman said: “The freezing temperatures have caused severe icing in some tunnels. Large icicles have damaged the overhead wires which is disrupting services between Glasgow Central and Gourock/Wemyss Bay.
She added: “It’s been a massive operation to clear almost 2,800 miles of track. ScotRail Alliance staff have worked day and night to clear deep snow and ice.”
In other parts of the country there are winds of up to 40mph which are causing snow to drift back on to tracks.”
Edinburgh Airport plans normal working today after 84 per cent of flights operated yesterday with 34 cancellations. Glasgow Airport warned passengers there may be more delays today. It operated 95 per cent of flights yesterday, with six cancelled.
Edinburgh Trams said it had carried 30,000 passengers since Wednesday.