Is it permissible to make one's wife swear by the Book of Allah that she will not use the new mobile phone in a way that displeases Allah, out of fear that she might use it for songs or similar things?

There is no religious impediment to a Muslim asking his Muslim brother to swear an oath to confirm or verify a matter, even if the accused is not a liar. Indeed, the righteous predecessors would administer oaths to one another. Examples include Ubaid al-Salmani administering an oath to Ali ibn Abi Talib (may Allah be pleased with him) regarding a hadith he heard from the Messenger of Allah (peace be upon him), and Umar ibn Abd al-Aziz administering an oath to Abu Burda ibn Abi Musa concerning a hadith he heard from his father, and Urwah ibn al-Zubair administering an oath. Therefore, there is no impediment to administering an oath to a wife regarding her not using a mobile phone or anything else for illicit purposes or for what is disliked.