Some people say that touching a neutral in this case will lead to the same result as touching the hot wire because it is AC and current path changes 100 times a second (we have 50 Hz.) Others say that the neutral wire will never shock you because it is firmly grounded at the transformer. Who is right?
Neutral is (somewhat) safe to touch when everything is operating correctly. As you noted, it is connected to ground so there shouldn't be any voltage difference between neutral and ground.
If there is a very heavy load (a lot of current flowing) and the neutral connection has a high resistance, then there can be a fairly high voltage on the neutral. High enough to hurt, maybe high enough to kill.
If the neutral is broken, then supposedly safe neutral will be connected to the hot through appliances connected to the outlets. That will be the full line voltage, certainly enough to kill.
In normal use, nothing (should) happen to you if you touch neutral. Something could be wired wrong or broken in the house, making it unsafe to touch the neutral.
Along with all that is also the fact that houses may be incorrectly wired. A wire that is color coded neutral (blue) may in fact actually be hot.
Normal practice before working on home electrical stuff is to check that the power is on at the outlet (using your meter or other electrical test device such as an electrician's screwdriver with a neon bulb) or whatever part you are working on. This proves to you that the wiring is live. It also shows you that your tools are working properly. At the same time, you can check live, neutral, and ground to be sure what voltage level is really present.
Once you know everything is powered, how it is wired, and that your tools are OK, you can switch off the power.
With the power off, check to see that your meter or test device truly shows that there is no voltage present on any of the lines you need to work on.
If this second test shows the power is truly off, then you intentionally connect the hot with the neutral. If things are really off, nothing will happen - else you get scary sparks and the circuit breaker trips.
When the power is really, truly, off you can get to work doing things that involve actually touching the wires.