What are the Sharia rulings and regulations for a woman's going out and returning, whether to the mosque or the market, and what are the evidences from the books of Hadith and Sira regarding women riding mounts during the time of the Prophet, peace and blessings be upon him?

When a woman goes out and returns, she must adhere to Islamic guidelines, including the Shar'i hijab with its conditions (not an adornment, opaque, loose, not perfumed, not resembling men's clothing or that of disbelieving women, and not a garment of fame). Furthermore, her going out should not involve unlawful mixing with non-mahram men, and it should be with her husband's permission, except for visiting parents according to some scholars. Her going out should also not lead to the neglect of her home, husband, and children's rights. The riding of animals by women was well-known during the time of the Prophet, peace and blessings be upon him, and there is evidence for this in the غزوات (battles/expeditions) and the Hijra (migration).