I think the line running from my house to the municipal sewer system has partially frozen. It's been below freezing here for weeks on end, with no end in sight. I hadn't noticed any problems until I ran a load of laundry today. After the washing machine drained, I found the bathtub was full of sudsy water that had obviously come from the washing machine. Eventually the water drained out, so I'm assuming the ice plug is only partially blocking the line. Should I try running hot water for a while?
I would start with boiling water - a couple of very large pots. Dump. If you get any kickback or notice that your water levels stay high repeat a few times. If you have ice or a blocked line, really hot water helps both.
Normally you would check your water levels in a cleanout or basement drain - from the sounds of it you have a bathtub at your low point or maybe it is at the same level. Usually I tell people to have one person look with flashlight on drain. If you see the water levels rise, even a little temporarily then usually there is some sort of blockage - I know that it may be ice in your case.
If the water isn't backing up into tub move on...
Then I would fill showers/sinks/tubs - anything that holds hot/scalding water and fill them until you run out - except for said tub that got the backwash. Again good to have another person at tub/drain. Dump everything at once. If it starts backwashing to tub then stop. You have to think just after a few times doing this that ice would be thoroughly melted and then you need to troubleshoot like your main line is clogged.
Hot water doesn't work. It doesn't melt as much ice as the ice can cool the water. Then there's cold water atop the ice preventing the hot from getting through.
Only works if there is movement in the line. But then again any temperature of moving water will clear ice.
Use the cleanout to determine where the blockage is. Apply heat.