Power cord without ground cable
Oscilloscopes are very useful pieces of equipment but they force us to tie our reference point for any measurement to earth because the chassis of the oscilloscope is tied to earth and so by extension the outer part of the BNC connectors for your test leads and several other user accessible parts.

Lets first consider why this may be a problem:

Case 1: You are testing a board that is powered from external bench power supplies (which we will assume have floating outputs) and you want to measure the gate drive across a particular FET gate-source. Here we will assume the source of this FET is connected to the local GND which is again floating from earth.

When we connect our scope lead return wire to the source of this FET we are connecting it to earth and we have a different path to earth via the capacitance in our bench supplies. This can lead earth current loops, even if AC only and increased noise in our measurement.

Case 2: You are trying to understand this simple switch mode PSU.



This is quite a simple circuit to understand but you will almost certainly want to look at the waveform across the IC drain-source which you can't easily do because the source of the device is not and can not be connected to earth.  If you try you will almost certainly destroy the unit under test and your scope.

What has been done by removing the earth connection is you have isolated your scope. This may work but you have created an unsafe scope because any conductive accessible parts on the will be at hazardous potential whenever you try to measure something that is referenced to this dangerous potential.
Even when looking at voltages that should be safe you have increased the risk of electric shock because in the event of the scope becoming faulty you have removed a vital safety protection mechanism.

This works because we are assuming the internal supplies required by the oscilloscope have suitable mains transformers to provide the isolation required but we can not be sure of this. A slightly less bad, though still horrendous, solution is to use an external isolation transformer.  This still has all the problems highlighted above but does at least give some indication all is not as it should be, and this equipment needs special safety precautions.

The Correct Way

For Case 1: Reducing noise may be as simple as wrapping your scope probe leads a few times through a ferrite ring to create a simple but often effective common mode choke. Alternatively you may be able to run your circuit from one or more batteries removing the capacitive earth path via the bench PSU.

For Case 2: Measuring the voltage across the IC drain-source 

a) If your scope has a math function then you can connect Ch1 to the drain, Ch2 to the source and look at Ch1-Ch2.  This is not perfect because if you need to look at two signals at once and you only have a two channel scope you're stumped.

b)  Use a suitable isolation transformer to isolate equipment under test, not the test equipment.

c) If isolating the equipment under test is impractical then you need a differential probe.  Good ones are expensive but at least they are safe.

Anybody caught floating a scope in my lab would be subject to disciplinary action.   



Edit Battery Scopes

There are two distinct types of battery powered scopes some are just like normal scopes except they are battery powered. These still have the property that any accessible connections are tied to the voltage of the signal return and so are fine for signals floating a few volts away from earth as long as that voltage is not hazardous.  Others have special differential inputs so may be used on higher voltage.

Another concern is interfaces. My scope has a USB output and so if I were to isolate my scope it would be connected back to earth as soon as I plugged the USB lead in.