How can I check if there is electricity in a power socket without a tester or any other specialized tools?
Note: The method should not involve the destruction of the socket or my whole power supply!
Plug in a working electrical appliance (like a phone charger) and see if it works.
There's no completely foolproof way to do this.
You can plug in a lamp or other appliance, but this is prone to false negatives (situations where the appliance won't work, but the outlet still has electricity). The most common situation would be a broken neutral wire (the hot wire is still live, but with nowhere for the electricity to go, the appliance won't work). More dangerously, a mis-wired socket with hot and ground switched won't produce a working appliance, but touching any exposed metal on the appliance can kill you.
If you want a definite answer, you need tools. In terms of increasing effort and accuracy, these are:
The hardcore test, still "safe". Only to be used if really needed.
It assumes that the voltage is standard 11V / 220V.
Use an thin wire, properly insulated at one end, NOT insulated at the other. Some screwdriver should do the job just fine.
Be VERY CAREFUL not to touch an metal part of the wire.
Insert the wire into one of the holes of the socket.
DANGEROUS: with the back of your hand "slap" the uninsulated part of the wire, in a way that will not make it possible to leave your hand in contact tot he wire. UNDER NO CIRCUMSTANCE SHOULD YOU TOUCH THE WIRE WITH THE PALM-SIDE OF YOUR HAND. THERE IS A RISK THAT YOUR HAND WILL CLENCH ON THE WIRE AND THAT WILL BE YOUR END.
DANGEROUS: If there is no shock, try again for a few times, very slightly extending the time of contact. If there is still no shock in the hand, the "hole" is not electrified.
Repeat 3-5 for the other hole.
It will work better if you wash your hands first, removing the excess water with a towel - espeially if your hands are usually very dry.