Electricians and power engineers know the implications of a disconnected or loose neutral conductor. But what about one of the hot conductor phases?
In three phase, for motors connected as delta, that means single phasing.
If they are connected wye (how common is this?), it could be different as there is some two dimensional aspect of the two remaining phases. So I might think that a motor wired this way and losing a phase could still start and run, in a derated manner.
One interesting issue I see with both three phase and single phase is the interaction between line-to-line loads (including delta connected three phase loads) and line-to-neutral loads. Looking at single phase for simplicity, all those L-L loads are not getting their 240 volts. But they are making a connection between the live line and the dead line, effectively energizing it at a high impedance. Then this voltage gets applied to all the L-N loads on the dead side. This results will obviously vary greatly depending on the mix and balance of L-L and L-N loads. But I see the potential for limited amounts of damage to some equipment attempting to operate on the low voltage and higher impedance.
Another interesting issue is probably only applicable to single phase. This is the metering. As I understand it, many, or maybe most, single phase energy meters for single phase just use a single transformer for each of current and potential. Both hot conductors pass through the current transformer, but in opposing directions due to their opposing phase angle. But if the loose hot conductor is loose ahead of the meter, the potential transformer will get a greatly reduced voltage reading, depending on the balance of loads connected to the dead phase and either neutral or the other phase. So whatever current does flow on the still live phase, it will be multiplied by the much lower (perhaps typically 1/4 as much) voltage the potential transformer reads. But I guess that is a blessing in disguise in a bad situation when the power company isn't delivering all phases.