voltage on Oscilloscope without signal applied
You are acting like a (very bad) antenna, picking up some of the EM field from the mains running in the walls around you.

The scope inputs are usually in the M\$\Omega\$ range, with a 10:1 you go up to 10M\$\Omega\$: such a great impedance means that when you connect something that can source even a tiny current, you see the corresponding waveform on screen.

Try to touch the probe tip and the probe ground, you will see that the sine wave is almost gone.

You do not get any shock because this current is very tiny. 240V peak on 10megs means  240uA, well below the 20/30mA that start to be dangerous, and somewhat below the "feeling" threshold.

I think that there's nothing wrong with your setup, I am quite surprised that your scope does not need earthing, every scope I have ever seen is connected to the mains with a three terminals plug. Using a three terminals plug can possibly mitigate this problem, anyway it is perfectly normal to see AC if you touch an high impedance terminal.