Why does one phase of my buildings mains line show a significant amount of voltage (~60V) when it goes down?
Most all residential power feeds in the USA are 120/240 Vac split-phase. You have two hot wires (L1, L2) and a neutral coming into your house or apartment or trailer. After passing through your power meter it goes to your breaker panel. It is constructed so 240 Vac appliances connect to L1 and L2 which are 180 deg out of phase. 120 Vac appliances connect to either L1 or L2 in a hop-scotch pattern that helps distribute and even out the loading placed on L1 and L2.

This also helps keep return currents in neutral small enough to be safe.

Because 240 Vac appliances literally pass current to/from L1 and L2, if L1 or L2 fails due to outside lines being cut or a burned out main breaker, the disrupted lines can have some voltage on it, known as backfeed.

To test for this condition just use the breakers to turn off all 240 Vac appliances. If the problem goes away then you have a burned-out main breaker (replace) or a phase wire from the utility is open (call utility service).