In a 24 hr period temps went from 32 to 3 degrees. Before I left I had a space heater running under the kitchen sink. Both hot and cold water were running fine and temps were in the teens. I came back in 7 hrs and the hot water ran full force but no cold running water. However, when I woke up in the morning with temps @ -5 the cold water now runs a steady thick trickle.

I have turned on both hot and cold to purposely trickle the water in hopes that it will melt any ice on the cold water side in the pipeline.

All my other faucets in the house run fine.

Is the cold water side just at thawing stage or is this a sign that something more disastrous could happen to the cold water pipe such as bursting.

How can I prevent the cold water pipe from bursting?

It may have an ice blockage or a small crack. Are you on a basement? Crawl space? Slab?

It is an older building on the 1st floor above the basement. I suspect any ice blockage would be in an interior pipe hidden because there is heat running under the sink.

Probably should apply some heat to pipes in basement. A heat tape on the pipes in basement close to the kitchen would probably be best and easy to do.