Is it permissible to build a small mosque in the basement of a house designated for residence, given that the plot of land belongs to the wife?

If the small mosque is a designated prayer area exclusively for the household, then there is no harm, as the predecessors used to designate prayer areas in their homes.

However, if it is a mosque open to the general Muslim public, the jurists have differed on the ruling of building residences above the mosque: The Shafi'is, Malikis, and Hanbalis permitted making the upper floor of a house a mosque without its lower floor, and vice versa. Abu Hanifa: If a mosque has a basement beneath it or a house above it, and the mosque's door opens to the street and is separated from his ownership, then it is not considered a mosque, and it is permissible to sell it or inherit it; because it has not been dedicated solely to Allah Almighty, unless the basement is for the mosque's benefits. It was narrated from Abu Hanifa that it is permissible to make the lower floor a mosque with a dwelling above it, but the reverse is not permissible. It was narrated from Muhammad the opposite of that. Abu Yusuf: Permitted it in both cases.

And what is considered the fatwa (definition) (prevailing legal opinion) with us is the permissibility of building residences above and below the mosque.