Northern tweeted: "Northern advise customers not to travel at this time in the Yorkshire area due to the severe infrastructure failure caused by lightening strikes."
LNER has also told passengers not to travel - causing many to vent their frustration.
Some passengers have been stranded on trains for more than two hours, while others have been stuck part-way through their journey.
HillsideSaint1988 tweeted that the "no travel" advice was useless, given some were already en route, adding: "What practically are we advised to do?
"Get off at Newcastle and get a hotel?"
Lines were blocked when lightening hit the Integrated Electronic Control Centre at York earlier this morning.
A spokeswoman for Network Rail said there would be disruption for the rest of the day.
She said: "Basically we are still in a situation where we have widespread severe disruption.
"We are getting some signalling equipment back on but it doesn't help when there has been such disruption to trains and crews."