(while I was writing this 10 other replies were posted; hopefully I am not repeating what others already wrote)
IME a differential probe is the method with no side effects to the oscilloscope and the DUT. Both equipments will have paths to ground and you can get shocked if you inadvertently touch the DUT in a single point. However, differential probes are somewhat expensive.
In this case, nothing prevents you from using an isolation transformer on the DUT and your brains. Isolating the DUT will not have a path to ground when left alone or connected to ungrounded equipment - useful when probing its innards with portable DMMs, for example. This minimizes the risk of electrical shock as it requires touching two points in the DUT.
The moment you probe the DUT with a grounded equipment you create a path to ground, which increases the risk of electrical shock as touching one point on the DUT is enough to cause the current to flow through DUT-->oscilloscope-->ground-->your body-->DUT. In this case the equipment itself is the one "anchoring" the potential to ground - thus touching its alligator clips or its chassis will not cause you any harm (provided you have a reliable ground connection, of course).
Isolating the oscilloscope increases the chances of you getting a shock as the path to ground is through the DUT and any alligator clips or the chassis of the oscilloscope will be elevated to the DUT potential. Also, touching a single point in the DUT is enough to cause the current to flow through DUT-->ground-->oscilloscope-->your body-->DUT.
Isolating both will help with your safety as there will be no path to ground at any point; this minimizes the risk of electrical shock as it requires touching two points in the DUT. Also, the DUT and the "grounded" test equipment will be at the same potential at all times, which will be a similar scenario as the portable DMM. However, one of the pitfalls of this method is falling prey to its advantages: it is easy to forget or assume the environment is safe and relax on procedure.
I suspect I didn't forget any details, but I am sure others will point out any flaws...