broken neutral conductor and the protective conductor

I presume one could use this for a broken neutral detection circuit. But short of this, I do not see any way one could reasonably and safely utilize this.

Always best to be both legal and safe. Therefore, the best answer is "no".

Do you mean using the possible potential difference between the Neutral and Earth (or Ground) due (generally) to inductive coupling, which in many (if not most) electrical systems is supposed to be bonded together and at the same potential to prevent an electric shock?

In what way and why on earth would you want to (this is totally impractical and extremely unsafe and in many circumstances can can lead to electrocution)? You cannot power an electrical appliance for free of this if that is what you are thinking!

Can you please provide more information, including your country (electrical systems vary from country to country).

Practically no. This voltage varies with imbalance currents elasewhere on the distribution network.

According to national standards, no. Such a concept is certainly outside BS7671.

Morally, no. In the event of a disconnection in the neutral upstream, the neutral-to-earth voltage rises to the full phase voltage. Therefore there is a risk of equipment damage, fire and electrocution, and therefore a risk of prosecution. No-one wants that, surely?

If in doubt, consult a qualified electrician.