Is it permissible for a mother to search her daughter's phone and view her private conversations that contain secrets and trusts belonging to her friends, and what is the proper way to advise her if that is not permissible?

A mother does not have the right to inspect her daughter's phone or read her private messages, as this constitutes spying, which is forbidden in Islamic law. It is impermissible except in certain specific circumstances. Children should gently and respectfully explain this to their mothers, showing them fatwas and the words of scholars regarding this matter.