My question is, since very recently we have found one or more mouse in our home, we can kill them by using mouse trap or poisoning them to get rid of them. So please tell me which way Allaah wants me to solve this problem.

All perfect praise be to Allaah, The Lord of the Worlds. I testify that there is none worthy of worship except Allaah, and that Muhammad is His slave and Messenger. We ask Allaah to exalt his mention as well as that of his family and all his companions.

Yes, it is permissible to kill the mice. The Prophet said: "There are five harmful creatures which should be killed even if one is in the state of Ihram: a snake, a speckled crow, a rat, a voracious dog, and a kite." The tools that should be used in killing the animal should be sharp in order not to torture the animal, as we are ordered in Islamic Law to kill the animal in a good manner. The Prophet said: "Verily, Allaah has prescribed Al-Ihsaan (i.e. performing an action in a perfect manner) towards everything. Thus, if you kill (anyone for some genuine reason), kill (him/her/it) humanely; when you slaughter (an animal), slaughter (it) well. And every one of you should sharpen his blade to avoid torturing the animal he slaughters." [Muslim]

Therefore, if the poison is strong and quickly causes the death of the animal, then it may be used to kill it. However, if the death of the animal is very slow, then it seems that it is not permissible to kill with such poison when it is possible to kill it with something else [causing less torture to the animal]. It is possible to use a mouse trap to catch the mouse and then kill it with any tool to kill the animal in a perfect manner.

Finally, it should be noted that there are some animals that we are not allowed to kill, like frogs and the like, so we should not kill them unless they are harmful. Imaam An-Nawawi reported that there is a consensus of scholars that it is permissible to kill harmful animals.

Allaah knows best.