With a hot chassis, always plug the hot chassis item into an isolation transformer. Then you are free to connect an oscilloscope or VTVM (both are typically referenced to earth ground) to the chassis without fear of destroying something.
You can buy isolation transformers or you can make your own by putting two identical transformers of decent power capability back to back. For instance, if you have a couple of identical "filament transformers" capable of 6.3 volts at 20 amps, you can connect the two 6.3v secondaries together, plug one transformer into the mains and use what was formerly the primary of the second transformer as an isolated secondary with the same voltage as the mains. In this case, you'll be capable of handling equipment up to around 120 watts. There will be some losses and inefficiency in the two transformers, so give yourself a bit of a cushion under that 120 watts.
Dean