Trying not to fry my oscilloscope

When I was much younger I had electronics as a hobby and had an old single channel all vacuum tube oscilloscope.
Never did I fear frying my scope by connecting the probe's ground clip in the wrong place.
After watching YouTube videos on the subject I now know that this can be a real issue.
My electronic brain cells has gone rusty over the years, so that makes me uncertain whether I really understand the message
these videos are trying to convey. The fact that the videos are made in countries with a different electrical system than my own
country doesn't make it easier.
So, I think I have gathered that a crude rule of thumb could go something like this:

Do not connect the probe's ground clip to anything other than a point in the circuit connected to earth ground.

Does this sound like a rule of thumb at all?

Another question:
Some videos mention using an isolation transformer where the secondary winding is NOT earth grounded in order to avoid frying an oscilloscope.
The Device Under Test must now be supplied from the isolation transformer and you're supposedly good to go.
But,-if the DUT is fed a signal from a signal generator which IS earth grounded, will the DUT also become connected to earth ground through the input connector of the DUT?
In such a scenario, should both the DUT and the signal generator be supplied from the isolation transformer?
 
With modern 2+ channel scopes you get a math function to mimic differential probing (not as good as a real differential probe, but certainly very handy).
Yes, a safe 'scope has earthed probe grounds, only connect these to mains earth referenced grounds or floating grounds, otherwise you'll have a something trip due to earth current, and maybe damage your circuit by shorting it out.

An isolation transformer is a great idea if working on any mains circuitry. Full isolation means you can get away with accidentally touching one point in the circuit (but not two), however you will want to ground the chassis to be able to probe, and this means you need to ensure the DUT chassis is definitely isolated from both live and neutral before starting(*).

Note a variac is not an isolation transformer. If in doubt check with a multimeter to see if a particular device is isolated (disconect it, turn any switches on it on, then measure from input to output live and neutral)

You can get x100 probes with decent shrouding which make using a 'scope on mains voltages much safer (make sure to get properly rated probes, not knock-offs).

If you are not working on the mains part of a DUT, ensure any mains stuff cannot be accidentally touched (cover with insulating panel perhaps, perspex is good if you want to see any lights or smoke(!)).

In general you never want your bench test equipment to be anything but earth grounded, so they are safe to operate without wearing PPE(!). Keep the risks to the DUT, isolate it if possible, and prefer differential measurement to redefining chassis ground. Power down before moving probes around, very easy to slip setting up a probe and that can lead to lots of damage.

Basically identify and minimize the risks, avoiding a risk completely is the best form of mitigation. Take time to do setup and do safety checks, know where the off switch is. Working alone is the most risky (but commonplace) situation, so stack the odds in your favour and never cut corners with electrical safety.

And one thing a lot of people forget, if a circuit carrys enough power/energy to be able spit out big sparks if shorted, eye protection is a wise move... That includes mains, power amps, big power supplies, large electrolytics. Electrons may not be able to jump through air by more than a few mm, but high current spark debris can, and can both burn and deliver a shock.

(*) some ancient equipment has live or neutral connected directly to the chassis, such are not safe to use or work on until this problem is fixed (3-wire mains lead, earth to chassis, disconnect other mains wires from chassis)
 
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Basically identify and minimize the risks, avoiding a risk completely is the best form of mitigation. Take time to do setup and do safety checks, know where the off switch is. Working alone is the most risky (but commonplace) situation, so stack the odds in your favour and never cut corners with electrical safety.
I used to service almost every vacuum tube guitar amp in town, and I was paranoid beyond belief.
Pulled the plug from the wall socket (didn't even trust the mains switch), hooked up the probes and applied mains voltage, not touching the probe/leads/multimeter etc.
After power down;-discharged the power supply with a high value bleeder resistor and let the bleeder resistor in place while doing soldering work.
Paranoid? -Yes, but I never got zapped!
An electrician I knew had got zapped so many times that the nerves in his fingers were long gone.
He checked for 230 volt mains by licking on two fingers and touched the terminals.
 
The videos I have seen make drawings and have lengthy explanations without really boiling it down to one simple sentence,
so here's another take on a crude rule of thumb:
Connect the ground clip of your probe anywhere you could safely poke you finger while standing barefoot on a wet concrete floor without getting zapped.
 
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With modern 2+ channel scopes you get a math function to mimic differential probing
You get that on the older analog scopes too. Even my TEK 2215 has invert and add functions. CH1+(-)CH2 = CH1-CH2. Handy indeed.

The issue with scope grounding is that the ground clip is connected to the frame ground of the scope. That, in turn, is connected to the building safety earth (i.e., the 'ground' pin in the mains power outlet). If the circuit you're working on is also connected to the mains safety earth and you connect the scope probe ground to a voltage other than the ground/earth reference of the circuit you will short circuit that voltage to ground. That draws a large current through the scope probe. Whether it'll fry the scope is another question. I doubt it will, but I also wouldn't go test the theory with my scope. Either way, it would result in a rather nasty surprise and flying sparks.

I've never found the need for an isolation transformer. I just connect the probe ground clip to circuit ground and move on. If I need to probe the voltage difference between two points that are both some voltage above ground I will use the CH1-CH2 trick as Mark mentioned.

Tom
 
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I use an isolation transformer so I can connect scope ground pretty much anywhere.

I bought it after blowing out all the house circuit breakers when I connected scope ground to negative output of a SMPS bridge rectifier !
When I was a kid I didn't need a scope to blow circuit breakers.
Strangely enough I knew that what was going on inside of wall outlets were no friend of fingers so I used pliers to poke nails into each hole and drop a third nail across the first two.

ElectricityTest.jpg
 
Do this if nothing else. Get a 10x probe. Use it without the ground to measure anyplace you may want to connect the ground clip too. Alternatively, use a DMM to measure between scope ground and the desired grounding point. If you see more than a few volts, that's not a place the clip can be connected too. Make sure all your outlets are wired properly.