Oscilloscope manufacturers have been screwing multitudes of noobs and overconfident lab rats with this same routine for decades. 
ALL oscilloscope inputs should be isolated from the mains, and channel to channel, and external trigger inputs, end of story
If that toplogy works for multimeters (and high end snob CRO/DSOs), why not oscilloscopes for the average home/student/repair tech punters?
It's DANGEROUS to flog a device that has it's single ended ground referenced to an earth linked neutral,
which in reality that so called 'neutral' is a grounded LIVE conductor with the same potential to go BANG as the Active line
Please spare us the 'extra cost of isolated inputs' BS, that's been flogged to death by apologists for years.
Mass production always sorts that out anyway, manufacturers and their reps are either DISHONEST and or have s*** for brains for continuing this practice for decades.
You would think with the release of cheap mass produced DSOs they would have done it by now... instead they flog expensive addons that can cost as much as the DSO itself,
or force desperate users to stuff about with half baked isolation transformers.
To the noobs reading this: WATCH YOUR BACK with standard CRO and DSOs, they are not 'floating' multimeters with a display
A good probe short BANG can send welding temperature metal shrapel into your eyes before you can react.
You won't read about that in any instruction manual, it might affect 'sales' and 'commissions' 
It's not an 'electronics learning experience' ordeal you need... 
In the past, most people using Oscilloscopes already had a fair degree of knowledge about the Mains supply that appears at the wall socket .(GPO in Oz speak)
(In fact, most people in the general population knew that you shouldn't mess with it!)
"Noobs" never got within a "bulls roar" of an Oscilloscope, without the mentoring of an experienced person.
After all, such an instrument was, in those days, anything from a years wages, to a new car, to a house, in terms of monetary value.
As they became used to working on equipment using an analog multimeter, they would become aware of the need for caution around Mains circuitry.
Most of the stuff they would be dealing with returned the metalwork to the Mains PE.
If the meter probe slipped, they could short the Active to chassis and Blam!
When they finally got to use 'scopes, they knew you couldn't put the ground clip on the wrong side of the Mains.
With linear power supplies, there was little reason to probe the primary side of the power transformer, apart from simply confirming the presence of Mains, which could be done without the ground clip connected to anything.( the PE of the 'scope provided the return path)
Another way was to use two probes & channels in "a+b"mode, with the ground clip either disconnected altogether, or connected to a known, verified, ground point.
Some (very few) Oscilloscopes were made with "floating" inputs---- my old BWD, for instance.
There are two ways of making such a scope, if it is intended to be powered from the Mains.
The first way is to connect both sides of the BNC connector to the input circuit via capacitors, with a normal "guts" with its common side connected to PE.
Unfortunately, that yields an "ac only" instrument.
The other way is to "float" all the internal circuitry.
The downside to this, is that, if you accidentally put the probe "ground clip" onto the Mains Active, all the "common" side of that circuitry will be at 230v RMS w.r.t earth.
All good, but that then places a requirement upon all of the rotary controls to have shafts insulated to that voltage or greater.
All the switches will have the same requirement.
Transformerless radios had the same problem, to an even greater extent, because one side of the circuitry was
always connected directly to one side of the Mains.
It was pure luck, which one!
Of course, my 'scope examples are more to do with conventional analog instruments, because they need such things as rotary controls.
The controls on DSOs may be a bit easier to isolate ( opto-isolators, etc).
If a 'scope has a metal case, that would normally be connected to protective earth in case some of the internal isolation fails over time.
For effective shielding, this may require capacitors between the internal "common" or the internal shielding & the case.
There we have some more potential points of failure.
It is a lot more complex than it seems at first glance.
Of course, anybody can get caught out, I have even been caught.
I always used an isolation transformer, but this day, someone had borrowed it, & I was quite gingerly, working without it.
Just for that second, I forgot, connected the ground clip & probe tip up to both sides of the Mains in a SMPS, walked over to the GPO, plugged it in & switched it on.
Bang! Nice flash, the RCD operated, & dropped off the supply to my workshop.
"Welding temperature shrapnel" ? ----Nah! Bit of a burn on the clip, not affecting its functionality.
(I have an oxy set, & have encountered welding temperature shrapnel when cutting sheet metal with some sort of grot on it)
You will note that I placed the scope probe connections while the DUT was unplugged & the GPO switch turned OFF.
I then took a few steps to plug in & turn on the power.
This is exactly what I did normally.
Once you have connected the probe, there's no need to hover over it, so any flash will tend not to be in your face.
If it is hard to clip the probe hook so it stays clipped, requiring you to hold the probe in position, check which side of the Mains is Active, using. DMM, beforehand.
I would urge you to try to clip it so it is hands free, though, if you at all can.
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