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, sallallaahu ‘alayhi wa sallam, is His slave and Messenger.
Anas narrated: “The Prophet never saved anything for himself for the next day.” [At-Tirmithi] However, the Prophet did not absolutely abandon saving. ‘Umar narrated: “The Prophet used to sell the harvesting of the date-trees of Beni Nadheer (a Jewish tribe) (which he, sallallaahu ‘alayhi wa sallam, gained as a war booty) and he would keep for his family what they would suffice them to eat for a whole year.” [Al-Bukhari and Muslim]
Ibn Daqeeq Al-‘Eed explained the two narrations saying that the first narration refers to saving for himself and the second narration refers to saving for his family even though he is benefitting from that as well. However, if he was lonely and did not have a family he would not have saved for himself at all.
The fact that the Prophet did not save for himself reflects a very strong state of faith as this shows the complete reliance on Allaah, but this is something that is not obligatory on every Muslim.
Therefore, it is permissible for you to save money for yourself as long as you fulfil the obligatory rights on you, like paying Zakah, spending on the people whom you are obliged to spend on, like the wife, children and parents and the like, within what is permissible without being stingy.
Saving money should not be in a bank that deals with Riba (interest and/or usury) because if you do so you are helping in Riba which is one of the grave major sins. Also, you should not take any interest on this saved money.
It should be noted that a Muslim should spend in charity and remember the saying of the Prophet : “Whom amongst you is the money of his heirs most beloved to him than his own money?” People replied: “There is no one of us who loves the money of others more than his own money.” Thereupon, the Prophet said: “Your money (that you benefit from in the Hereafter) is that which you give in charity and the money of your heirs is what you save.” [Al-Bukhari]
For more benefit, please refer to Fatwa 92234.
Allaah Knows best.