If it is not connected and a scope probe's shield is connected to a dangerous voltage (e.g. mains), then then entire scope's case will be at that dangerous voltage.
True, but if it were earthed it would probably just blow up your probe or scope.

You shouldn't be attaching your input/probe ground to anything other than a grounded point on your device under test or bad things are bound to happen.
It is also possible, depending on why the scope was "floated", that there is internal damage to the mains transformer.
No, it was designed to be floating, just like most older scopes I've seen, especially the old vacuum tube based ones intended for the TV repair industry.
It appears to have a metal case that is connected to the probe shields, which is normal for a scope like this. Such instruments must be earthed - for other people's safety[1] and to prevent other equipment being damaged.
Hyperbole much?

You're more likely to damage your scope or your DUT if your scope is grounded. Don't get me wrong, I would normally run with it grounded too, but it actually makes equipment damage
more likely, not less. It reduces the chance of you shocking yourself if you do something silly but not the chance of equipment damage. If you're probing a device where the spot that you put your measurement ground is at a voltage that will give you a shock, you should be running your DUT on an isolation transformer.
What do you mean "constructed floated"? Do you mean that:
- the original documentation states it is designed to be operated when floated. But you don't have that information
- you have examined the interior circuit, and it is clear that it has never had a protective mains earth and never needed one. But you are a beginner, and so could not tell
He has the schematic. It shows that it is floating. He could probably earth it using the banana jack on the back (assuming it is reasonably well connected to the chassis, not just connected to the transformer shield like it shows in the schematic) if he chooses (my Stark has no such provision, BTW...) but it
came from the factory this way, it hasn't been modified.
He could change the input cord to permanently ground the chassis but that really doesn't make sticking your finger or probe ground to anything but ground any safer and removes the ability of sticking the scope ground on, say, a 12v power rail, to reference to that for measurements. Having your scope chassis at 12v isn't dangerous, you just need to know what you're doing.

If you want to be safest, use a GFCI upstream of the scope and ground the scope frame. Then your GFCI will trip (hopefully) preventing any damage to anything should your probe ground go astray.
