Purpose of isolation transformers

[dasein]

What is the point of isolation transformers? Had the idea to try to diy one in my head for awhile after watching the how to not blow up your oscilloscope () if I ever managed to find two identical transformers. I am now in possession of two identical transformers (hefty ones from backup battery ups,) but it seems like plugging test equipment into an isolation transformer is a bad idea? If it is a bad idea to plug the oscilloscope into an isolation transformer and a bad idea to plug test equipment into it, what exactly is the point of this piece of equipment at all?

Example cases:
- Using oscilloscope to test a circuit powered by wall wart that has an earth connection (plugging wall wart into isolation transformer).
- Using oscilloscope to test an arduino powered by a charging laptop (plugging laptop charger into isolation trasnsformer).
- Using oscilloscope to examine the DAC in powered benchtop multimeter (plugging multimeter into isolation transformers).

Making a post specifically because most other places I have looked have been focused on the safety aspect and just encouraging the person making the post to not mess with mains voltage or use an isolation transformer if they don't know what they are doing. Obviously safety is a huge concern and I am aware of that and that is why I am making this post. If I still don't find that I have a justifiable reason or sufficient knowledge to own much less build one, then I won't. Going down this rabbit hole though I really just want to know more about why these things even exist in the first place. Thanks.

[DonKu]

See if this video helps:

[bdunham7]

Going down this rabbit hole though I really just want to know more about why these things even exist in the first place. Thanks.

Obviously "isolation transformers" (which is almost all transformers except autotransformers) have been around forever.  In the context of using a 1:1 transformer just for isolation on the bench as one typically imagines, they were originally marketed as a service accessory for working on TVs and radios that had a live chassis.  The purpose was to mitigate the signficant shock hazard that results from having a large chunk of exposed metal at a dangerous potential. 

[wasedadoc]

Going down this rabbit hole though I really just want to know more about why these things even exist in the first place. Thanks.

.... The purpose was to mitigate the signficant shock hazard that results from having a large chunk of exposed metal at a dangerous potential

with respect to ground and able to source (to ground) sufficiently high current to be dangerous to human life.

In the same vein, shaver sockets in UK bathrooms must have an isolation transformer.  That a cheap to add tap on the secondary winding can provide 110 Volts for visitors from some other parts of the world is a bonus, not the primary reason for the transformer.
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[dasein]

Going down this rabbit hole though I really just want to know more about why these things even exist in the first place. Thanks.

they were originally marketed as a service accessory for working on TVs and radios that had a live chassis.  The purpose was to mitigate the signficant shock hazard that results from having a large chunk of exposed metal at a dangerous potential.

How does this translate to modern test equipment such as a benchtop multimeter? In these cases I was under the impression that earth ground was to prevent a floating ground and therefore a live chassis in the case of a fault...

[Wallace Gasiewicz]

As described in the previous posts. it is primarily a safety function.   

It can be thought of in the same way as an "Isolated DC Power Supply" in that neither pos or neg is grounded (Earthed). Obviously in AC the Plus and Minus are changing at 50 or 60 Hz.
As Bdunham7 already pointed out.   If all the power is run thru a transformer, the power is essentially Floating or Isolated.   Until you "reference" the Neg to GND.   
Lab PS (DC and AC) use this arrangement. The Chassis can be grounded but neither pos or neg is referenced to GND.   
There are implications both good and bad.   Running everything thru an Isolation Transformer, the chassis is not Grounded.

In the US there were many radios that had no transformer and one end of the only two wire AC was attached to chassis. Also the two wire plugs were symmetrical.  US at that time 110 V AC.   I think a popular radio like this was an "All American 5"  (5 valves), the heaters of the valves were also fed in series with the 110 AC.  They could be a problem especially if operated on the kitchen sink.
Today, Oscilloscopes almost all have their Neg connected to GND, there are implications if you "float" a scope.
I think Benchtop Multimeters like HP 3456 have "Floating Inputs".  Handheld Multimeters are floating because of the battery power.

[bdunham7]

How does this translate to modern test equipment such as a benchtop multimeter? In these cases I was under the impression that earth ground was to prevent a floating ground and therefore a live chassis in the case of a fault...

I'm not sure how your two sentences are related.

A modern bench multimeter will be "guarded" and the "in-guard" region--which includes all  the inputs--will be isolated from anything else including ground.  That is, except for parasitic capacitive coupling and resistive leakage.  The "out-guard" region, if it has one, will be referenced to ground.  There will be an isolation system between the two. 

An earth ground does prevent the chassis from floating and that is an entirely separate function.  Removing the earth reference from the mains power with an isolation transformer is not directly related to breaking the protective earth ground connection.  You can do either or both, except doing the latter without the former is probably a bad idea.