I realized from the OP that there are other “infrastructures” that haven’t been covered.
Overhead wires (electricity, telephone, cable TV, etc.): Not much problem from extreme cold; in fact the cold reduces resistance and makes the electrical wires more efficient. Once in a while, the shrinking from the cold will actually pull connections apart, but this only happens if the installers made a mistake in calculating the amount of slack needed in the wires. That’s pretty rare.
There are obvious problems with ice storms, which knock down trees onto wires, or just add enough weight to wires to pull them down. This isn’t really a problem of the extreme cold, in fact, these are more a problem when the weather is warmer.
There are occasional problems when an entry point (electrical meter, phone junction box, etc.) on the side of a house is damaged by a falling icicle. But again, this isn’t caused by the cold; usually it’s the melting when spring comes.
Gas lines: these are almost all buried, and the gas in them doesn’t freeze, even at these temps. Occasionally contamination (usually water) gets into the gas lines, and can freeze, since they may not be buried as deep. But that’s unusual.
Streets: well, the temp changes & water thawing/freezing end up causing potholes in streets. But that’s basically just repair work to be done; it doesn’t prevent the street from being used.
There is ‘frost heave’, which can damage a street enough to make it impassible. But in a normal city grid, there are other streets that can be used with a slight detour, so this has a minor effect on the city infrastructure as a whole. Also, this happens mostly during Fall & Spring, when ground is freezing & thawing.
There is the obvious problem of snow removal & dealing with ice, but we have a lot of experience with snowplows & snowblowers, and with using sand & salt on roadways.
District Heat/Cooling systems: many cities have district heating (and cooling) systems in the downtown areas. These generally use underground tunnels to provide heating to various buildings. In the winter, this is heated air or water being circulated to these buildings, so it doesn’t have problems with freezing.
Railroad tracks: they can have problems with freezing switches, as Broomstick mentioned. And extreme cold will occasionally shrink rails enough to leave a gap, thus making the ride rougher. But it’s pretty rare that this is enough to make the track impassible.
**Tunnels:**Here in Minnesota, we recently had problems with ice inside tunnels! Under the Airport & Minnehaha Creek, water seeped into the light rail tunnel. This was not expected to be a problem, they had included drains for this, and it was enough underground that it would stay liquid. However, near the entrance to the tunnels, the incoming trains pulled in enough cold air that this water froze, and made the tracks slippery, which caused traction problems for the railcars. I believe they fixed this by installing heaters to keep the water liquid until it drained away. There have been similar problems in some freeway tunnels.
Sidewalks: Snow & ice removal is more of a problem on sidewalks, where there are always some people who won’t bother to clean their sidewalks. If it gets bad enough, the city will send workers out to do that, and add the cost to their property tax bill.
I think this covers most of the city infrastructure.