MOSFET on Arduino Uno
Your selection of transistor IRF520 if ok. The required gate threshold is 4 volts or less, as seen on data sheet seen here : Data sheet  Your Arduino is putting out 5 volts. 

The drain load of your device seems ok with being a 12K equivalent and drawing only 1mA at 12 volts is also ok. The IRF510 will turn off even with a high resistance device in the drain circuit. 

Your gate resistor is also ok.

Replace your load device with a 12K resistor and test again. You have shown that your transistor will be off when you disconnect the Arduino pin from the gate (resistor pulls gate low).

Since this is a DC circuit, use voltmeter rather than oscilloscope. The problem is in the measuring and possible ground conflicts. Make sure that your 12 volt supply ground has a good connection to the Arduino ground.