Use your DMM to check for a voltage between the ground clip of the scope, and the Arduino ground, with everything else connected (except the scope) as when you blew the LED.
The "ground side of the LED" (cathode) on the Arduino plain Uno is connected to the Arduino common ground and the anode of the LED is connected via a pair of resistors to the Pin 19 output of the ATMega, and to the Pin 13 Arduino output connector, and also to the Pin 3 of the ICSP connector.
The Uno R3 is slightly different, with the anode of the LED connected (through a resistor) also to an op-amp, the non-inverting input of which is connected to the Pin 19 of the ATmega 328. Cathode again connects directly to the common Ground.
I can't figure out how connecting the scope ground to the _ground_ side of the LED could blow the LED, even if there is a voltage present between the scope ground and the Arduino ground, if nothing else is connected.
Are you quite sure you connected to the Cathode (ground) side of the LED? It is the side with the green marking on it.