I have AC/DC 3/6/12V out universal adapter with two pins (without earth connector) and would like to use it for powering my arduino project and playing with oscilloscope. Knowing that my oscilloscope will make big bang if power supply is connected to main earth ground, I would like to test somehow if some of the AC 220V wires are connected to main earth ground. I know that my AC/DC is safe because of plug type but how to know that somewhere in the system main earth ground is not connected to phase or neutral.
The ground of your oscilloscope will be connected to mains earth assuming you're using a socket that has an eart connection. For safety, it should have this !
If the mains adapter you're using to feed your arduino board has a main earth connection then the same will aply tho this adapter.
Most mains adapter however do not have a mains earth connection and these will be floating with respect to earth.
For both adapter cases it is safe to connect the arduino's ground to the scope's ground.
To safely measure mains voltage on a scope you need to use an isolated probe or measure through a transformer for isolation.
Some (battery operated) portable scopes have an isolated ground and this in combination with appropriate high voltage rated inputs is enough to measure mains directly. But most bench scopes are unsuited for this.
I also highly recommend this video by Dave from the EEVBLOG !
If you use your AC/DC adapter to power your arduino board, and use the oscilloscope just to look at signals on the board then there will be no issue with your oscilloscope blowing up. The 3V/6V/12V power output from the adapter is not connected to ground in any way, it is isolated or "floating". The problem can arise if you use your oscilloscope to connect to the 220V AC voltage. Then you can expect a big bang. There are ways around this too, but it's safer not to do it!
Figure 1. Isolated power supply. Earthed oscilloscope.
This simplified circuit shows that the low-voltage DC supply is electrically isolated from the mains. You can safely connect either V+ or V- (but not both together) to the oscilloscope ground.
My old analog scope has a transformer to supply CRT and all analog circuits, I know beacause I got a schematics with it. So, whenever I need to measure like you, I do connect a mains plug that has the earth pin removed, now I get a floating case and scope's ground. It shall be noted that I use gloves and watch the other people if they move around the scope, since it can be lethaly dangerous, also it is advisable to know or measure a priori, the signal that will be connected to chasis with probe's croco clips, like ground, neutral, earth, not a live wire or +DC bus.