The Prophet (peace and blessings be upon him) commanding Abu Bakr (may Allah be pleased with him) to lead the people in prayer

It is narrated from Sayyiduna Sahl bin Sa'd (may Allah be pleased with him); he said: Bilal (may Allah be pleased with him) gave the call to prayer (adhan) for the Zuhr prayer. Meanwhile, a clamour came from the direction of Banu Amr bin Awf that a quarrel had broken out among them, to the point that they had thrown stones at one another. So the Prophet (peace and blessings be upon him) went to them (to make peace) and said: "O Abu Bakr, if the prayer is started, go forward and lead the people in prayer." He said: "Very well."
This hadith is sahih (authentic) on the condition of the two Shaykhs (Bukhari and Muslim), but they did not record it. in this form; what they agreed upon is that the Prophet (peace and blessings be upon him) said this during the illness in which he passed away.

It is narrated from Sayyiduna Anas bin Malik (may Allah be pleased with him); he said: Banu al-Mustaliq sent me to the Messenger of Allah (peace and blessings be upon him) and said: "Ask the Messenger of Allah (peace and blessings be upon him) for us: to whom should we hand over our sadaqat (zakah) after you?" I came to him (peace and blessings be upon him) and asked, and he (peace and blessings be upon him) said: "To Abu Bakr." I went back to them and told them. They said: "Go back to him (peace and blessings be upon him) and ask: if something (death) befalls Abu Bakr, then to whom?" I went and asked, and he said: "To Umar." They said: "Go again and ask: if something befalls Umar, then to whom?" I went and asked, and he said: "To Uthman." I went back and told them, and they said: "Go and ask: if something befalls Uthman, then to whom?" I went and asked, and he (peace and blessings be upon him) said: "If something befalls Uthman, then for you there is nothing but ruin upon ruin until the end of time."
This hadith is sahih (authentic) in its chain of narration, but the two Shaykhs (Bukhari and Muslim) did not record it.

It is narrated from Sayyiduna Ibn Umar (may Allah be pleased with them both) that the Messenger of Allah (peace and blessings be upon him) said to Abu Bakr and Umar (may Allah be pleased with them both): "After me, no one will rule over the two of you."
This hadith is sahih (authentic) in its chain of narration, but the two Shaykhs (Bukhari and Muslim) did not record it.

It is narrated from Habib bin Habib; he said: I was present with the Messenger of Allah (peace and blessings be upon him) when he (peace and blessings be upon him) said to Hassan bin Thabit (may Allah be pleased with him): "Have you said anything about Abu Bakr? Recite it so that I may hear." He said: (I have composed these verses:) "He was the second of the two in the lofty cave, while the enemy, having climbed the mountain, was circling around it; and he was the beloved of the Messenger of Allah (peace and blessings be upon him); people know well that among all creation he (peace and blessings be upon him) held no one equal to him."

It is narrated from Murrah al-Tayyib; he said: Abu Sufyan bin Harb came to Sayyiduna Ali bin Abi Talib (may Allah be pleased with him) and said: "What is the matter that this affair (the caliphate) has gone to the smallest and weakest clan of Quraysh (meaning Abu Bakr)? By Allah, if I wished, I would fill this place against him with horses and men." Sayyiduna Ali (may Allah be pleased with him) said: "O Abu Sufyan, for a long time you have been hostile to Islam and its people, yet that has not harmed Islam in the least. Indeed, we have found Abu Bakr worthy of it (the caliphate)."