ScotRail has advised customers to expect continued disruption of services.
Teams of Network Rail engineers have been checking routes through the night since Friday and carrying out repair work, with around 500 individual incidents reported.
These include more than 120 trees on tracks, damaged overhead wires, power supply failures and other objects and debris blown onto the line.
David Ross, ScotRail communications director told Radio’s Good Morning Scotland programme it had been “a very challenging weekend” and there was “still significant impact across the country”.
He said: “A lot of work has been carried out over the course of the weekend by colleagues at Network Rail.
“The vast majority routes are up and running but there is still ongoing disruption is some parts of the country.”
He said damage to overhead wires had caused cancellations in Ayrshire and Inverclyde. Overhanging trees at Whifflet in Coatbridge, had also caused disruption and trees were still falling onto the lines elsewhere in Lanarkshire.
“It is a changing picture and it will change over the course of the morning again,” Mr Ross added.
He warned some damage won’t be a “quick fix” and disruption could last a few days.
