The transformer issue is a tough one. Off-line powered instruments rely on the 50/60 Hz or switching power transformer for isolation from neutral and hot (1) and floating the instrument by breaking the ground connection increases the potential across the transformer. Some old instruments specified a peak float voltage and incorporated a special transformer to handle floating operation.
During floating operation, any exposed ground connection including the BNC connectors can become a hazard. Instruments designed with this in mind have special construction which tries to prevent exposure and include special BNC connectors and probes. Fluke uses or used plastic BNC connectors much like the shielded banana jacks now common on multimeters with probes to match; none of the BNC coaxial ground connection is exposed.
Even if you do float the oscilloscope safely using an external isolation transformer, considerable common mode capacitance still exists and all of the probe grounds are still tied together. Oscilloscopes designed with isolated inputs largely avoid these problems although their inputs are still unbalanced by common mode capacitance through the environment and the isolated power supply.
If you read documentation from scope manufacturers back then, they also recommended against it.
It depends on the time period and manufacturer. Some recommended floating the oscilloscope by disconnecting the ground and even specified the maximum floating voltage with little or no warning about the dangers. Suitable high voltage differential probes were very rare (2) and isolated probes and inputs did not exist leaving A-B measurements with their limitations, floating the device under test which is not always practical, and floating the oscilloscope.
Unlike in a formal course where participants have some understanding of the circuitry and the risks involved, on a public forum where all and any can read and deduce rightly or wrongly the procedures and risks involved it is nothing but irresponsible to discuss the merits of floating a scope. Sure some do it but it should never be promoted in a venue such as this as we have no idea of the skill level and understanding of those reading.
IMNSHO the only way that statement could have been improved is if it had been posted in a larger font.
You can both bite me.
Do you really think it better to not discuss the merits and hazards and instead rely on an argument from authority or silence? How do you expect people to gain understanding without discussion? Nancy Reagan's "just say no" campaign was as much a success as abstinence only education.
If your neutral/earth are tied together you need to revise your wiring.
In the US, split phase 240 volts AC comes down from the power pole and neutral is bonded to earth ground at the main distribution panel. One utility transformer supplies multiple homes leading to multiple grounds but it seems to work well enough despite multiple distributed earth grounds; this might even be preferable in areas with thunderstorms.
(1) Some already already isolated instruments include a second isolation barrier. Oscilloscopes with separate isolated inputs are the obvious example but some multimeters include a separate isolated supply for their input circuitry. This provides a higher isolation rating than a common or even special 50/60 Hz or switching power transformer will support and with a lower common mode capacitance.
For instance the isolated outputs of the Tektronix TM500 series power supplies are limited by the isolated windings on the 50/60 Hz power transformer in the mainframe to a floating voltage of about 50 volts. But the multimeters which use the same TM500 mainframes have their own internal isolated power supply allowing their common inputs to operate at up to 1000 volts.
(2) Tektronix had some suitable differential amplifiers starting in 1960 but the first suitable x10 probes which could be used with them in a high voltage application were introduced in 1962. They introduced a good differential amplifier in 1967.