Hadith: No Zakah is to be paid by the Muslim on his slave or horse

عن أبي هريرة رضي الله عنه أن رسول الله صلى الله عليه وسلم قال: «ليس على المُسلِم في عبدِهِ وَلاَ فَرَسهِ صَدَقَة». وفي لفظ: «إلا زكاة الفِطر في الرقيق».
[صحيح] - [متفق عليه]

Abu Hurayrah (may Allah be pleased with him) reported that the Prophet (may Allah's peace and blessings be upon him) said: “No Zakah is to be paid by the Muslim on his slave or horse." Another wording: "Except Zakat al-Fitr that is to be paid on one's slave.”
[Authentic hadith] - [Narrated by Bukhari & Muslim]

Explanation

Zakah is founded on justice and equality, hence, Allah has made it obligatory upon all types of properties, growing or apt to growth, such as harvested crops and merchandise for sale. As for properties that do not grow, those intended for possession or personal use, no Zakah is due on its owners, because it is a personal item. This category includes what one rides, be that a horse, a camel, or a car. The same rule applies to the slave used for service, one's furniture and utensils made for personal use. However, the exception here is Zakat al-Fitr on the slave, which is obligatory because it is associated with people not properties.