I'm working on an electronic home experimental project. But I have always been afraid of the issue of electricity since it can electrocute.
There are indeed dangers to working with electricity. To be clear, "electrocute" means specifically that you are killed(not injured) by electricity.
My question is.. Am I safe working with a power supplier of 12 DC connected to the AC wall socket?
Specifically from the danger of current passing through you, 12VDC is relatively safe provided you work carefully and dry. Unless you have broken or wet skin or provide an unusually low resistance to the source you're unlikely to pass dangerous currents through yourself, but just how safe you are will depend on reasonable precautions and the quality of design of your power supply. If it fails(or you miswire it) in such a way as to apply 120/240VAC to yourself it could easily stop your heart.
In addition to this, there are other dangers, shrapnel/toxic fumes from exploding parts, arc flash, ignition sources, inductance and capacitance. Inductors can produce higher voltages than input voltage when presented with high impedance, and capacitors can store dangerous amounts of energy for substantial periods of time after a circuit is deenergised.
12V@10A puts 120W at your disposal, so there are things that you could set up improperly that might hurt you. If you built/set up a current source boost converter wrong, the input might be 12V but if you connected the output with your body, 120W(VA) could allow it to run a substantial current through you. Follow known safe procedures, and try to limit yourself to building designs you can understand as much as possible.
Taking into consideration that the wires of the DC supplier will be exposed to feed my circuit. I have worked previously in this way with 5 volt power suppliers to feed digital circuits.
Don't have much to say here, but note that if you believe the exposure of a set of wires might present a substantial danger (like that they could be easily shorted by a dropped object or metal flyings and arc or spark) it's better to simply insulate them than to wonder.
But still my uncertainty remains, for example if the transformer AC to DC(power supplier) does not work then AC could be dangerous if I touch the exposed wires of the DC supplier?
I think you mean "If a transformer based AC->DC converter fails, could it pass the more dangerous AC voltage through to the DC output" and as much as possible I would imagine reputable supplier attempt to design consumer products to fail safe, but be aware that a component part, like a car part, or some cheap voltage converter board from aliexpress might not include every safety feature that is relevant to your application.
The power supplier that I want to use is 12 VDC with 10A of capacity.
Thanks a lot for your answers,
But what to do, what to do?
Wear PPE. Safety glasses or a full facemask, tinted or no, special laser goggles, gloves, whatever you need to wear to take as much skill as possible out of the safety equation.
Don't work live, and energise circuits from a safe location. When you must work live, have a clear and complete plan before you begin the work. Avoid using Power supplies that are unnecessarily powerful for repeated live work (I imagine almost everyone pops a wire around or adds a resistor on a 5V breadboard now and then), so if you know you're going to just mess with 5V logic circuits for a few months, it's unlikely something will happen, but a 5V 2A supply is probably adequate for most projects, and is much less likely to produce a significant arc/spark/hot surface ignition source than a 5V 200A supply (which you might have heard referred to as a welder). Extreme example, but the principle is sound at less extreme levels.
If you are adjusting live 5V components, there is a temptation to lean in, obtain a better view of what you're doing. This can increase your exposure to flying particles, and particularly, released gases. When you catch yourself doing this, relax, sit up straight, get your face out of the danger zone. You don't want to breathe capacitor smoke or LED smoke. Yuk. Look at the situation. Should you add/move a light source? Do you need a Helping Hands with a magnifier or a dentists lens?
Greetings. It's quite odd to end with greetings, so I started my reply with a farewell.