Short answer
Not as a specific sunnah. There is no authentic hadith that the Prophet ﷺ owned a cat or told Muslims to keep one, and the famous story of his cat Muezza has no reliable source. Keeping a cat is permitted and cats are pure, and caring for any animal, including feeding stray cats, is rewarded.
Keeping a cat is mubah (permitted), and kindness to it is rewarded. Cats are pure, and their leftover water is pure but mildly disliked (makruh tanzihi) for wudu when other water is available. There is no sunnah act of owning a cat.
The four schools agree on this.
Explanation
Cats have a warm place in Muslim culture, and the sunnah gives real reasons for it. A cat came to drink from Abu Qatada's water and he tilted the vessel for it, quoting the Prophet ﷺ: cats are not impure, they are among those who go around among you. The Companion known to us as Abu Hurayrah earned that nickname, 'father of the kitten', because he used to carry a small cat with him while herding.
But many popular stories go beyond the evidence. The tale that the Prophet ﷺ had a cat named Muezza, and cut off his sleeve rather than wake her, is not found in any reliable collection of hadith. Scholars of hadith treat it as baseless. It is better to share what is authentic.
So owning a cat is permitted, not a sunnah you are rewarded for in itself. The reward comes from how you treat it. The Prophet ﷺ told of a man forgiven for giving water to a thirsty dog, and said there is a reward in serving every living creature. He also told of a woman punished in the Fire because she locked up a cat and neither fed it nor let it find food.
What earns reward:
- feeding, watering and sheltering your cat and getting it treated when it is ill;
- feeding stray cats, putting out water in hot weather and helping injured animals;
- adopting a cat that needs a home, if you can care for it.
For keeping pets in general, see can Muslims have cats, and more history in Cats in Islam. The four schools agree on this ruling.
The evidence
Quran, Surah Al-An'aam 6:38
وَمَا مِن دَآبَّةٍۢ فِى ٱلْأَرْضِ وَلَا طَٰٓئِرٍۢ يَطِيرُ بِجَنَاحَيْهِ إِلَّآ أُمَمٌ أَمْثَالُكُم ۚ مَّا فَرَّطْنَا فِى ٱلْكِتَٰبِ مِن شَىْءٍۢ ۚ ثُمَّ إِلَىٰ رَبِّهِمْ يُحْشَرُونَ ﴿٣٨﴾
And there is no creature on [or within] the earth or bird that flies with its wings except [that they are] communities like you. We have not neglected in the Register a thing. Then unto their Lord they will be gathered.
Animals are communities created by Allah and deserve kindness.
Sunan Abi Dawud 75
Graded hasan sahih... إنها ليست بنجس إنها من الطوافين عليكم ...
"Cats are not impure; they are among those who go around among you."
Cats are pure and part of daily household life.
Jami' at-Tirmidhi 3840
Grading to be checked... كنت أرعى غنم أهلي فكانت لي هرة صغيرة ...
"Abu Hurayrah said: I used to herd my family's sheep, and I had a little cat."
The origin of Abu Hurayrah's nickname.
Sahih al-Bukhari 2363
Sahih... في كل كبد رطبة أجر ...
"There is a reward for serving every living creature."
Caring for cats and strays is rewarded.
Sahih al-Bukhari 3482
Sahih... عذبت امرأة في هرة سجنتها حتى ماتت ...
"A woman was punished because of a cat she locked up until it died."
Neglecting an animal is a sin.
Where the ruling is recorded
- Radd al-Muhtar (Ibn Abidin), Kitab al-Taharah, chapter on water, section on leftover water (su'r). A cat's leftover water is pure but mildly disliked for wudu when other water is available.
Checked by our team. For a more accurate answer to your own situation, consult a scholar. Quran translation by Sahih International. Hadith Arabic from the standard editions of the six books and the Muwatta, with gradings as recorded there; the English is our own rendering. Spotted a mistake? Tell us.