Did Allah Speak Directly to Prophet Muhammad?
Question

What is the evidence that Allah, may He be Exalted, spoke directly to Prophet Muhammad (blessings and peace of Allah be upon him)?
Praise be to Allah.The ways Allah communicated with His Prophets
In His holy Book, Allah, may He be Exalted, has told us of the ways in which He sent revelation to His Messengers (peace be upon him), one of which was speaking from behind a veil. Allah, may He be Exalted, says (application of the meaning): {And it is not for any human being that Allah should speak to him except by revelation or from behind a veil or that He sends a messenger [i.e., angel] to reveal, by His permission, what He wills. Indeed, He is Most High and Wise} [Ash-Shura 42:51]
He, may He be Exalted, stated that speaking from behind a veil is indicative of the lofty status of the Prophet who is spoken to, as He, may He be Exalted, says (interpretation of the meaning): {Those messengers - some of them We caused to exceed others. Among them were those to whom Allah spoke, and He raised some of them in degree} [Al-Baqarah 2:253]
The Messengers spoken to by Allah
Among those who were spoken to directly are:

Adam (peace be upon him)

It was narrated from Abu Umamah (may Allah be pleased with him) that a man said: O Messenger of Allah, was Adam a Prophet? He said: “Yes, one who was spoken to.” He said: How long was there between him and Nuh? He said: “Ten generations.” (Narrated by Ibn Hibban in his Sahih (14/69); classed as authentic by Shu`ayb Al-Arna’ut, the commentator on the book)

Musa (peace be upon him)

Allah, may He be Exalted, says (interpretation of the meaning): {And Allah spoke to Moses with [direct] speech} [An-Nisa’ 4:164]; {And when Moses arrived at Our appointed time and his Lord spoke to him…} [Al-A`raf 7:143]; {[Allah] said, “O Moses, I have chosen you over the people with My messages and My words [to you]”} [Al-A`raf 7:144]

Muhammad (blessings and peace of Allah be upon him)

It is proven in the story of the Prophet’s ascent (Mi`raj) to heaven, concerning which he (blessings and peace of Allah be upon him) said: “… Then I was taken up and came to Musa. He said: What has been enjoined upon you? I said: Fifty prayers every day. He said: Your Ummah will not be able to do fifty prayers every day. By Allah, I tested the people before you and I tried hard with the Children of Israel. Go back to your Lord and ask Him to reduce it for your Ummah. So I went back and He reduced it by ten for me. Then I went back to Musa and he said something similar… Go back to your Lord and ask Him to reduce it for your Ummah. I said: I have asked my Lord so much that I feel too shy. Rather I shall accept and surrender. (Narrated by Al-Bukhari (3674) and Muslim (162)
Al-Hafidh Ibn Hajar (may Allah have mercy on him) said:
“This is among the strongest evidence that Allah, may He be Glorified and Exalted, spoke directly to his Prophet Muhammad (blessings and peace of Allah be upon him) on the night of the Isra’ (night journey) without any intermediary.” (Fat-h Al-Bari, 7/216)
Ibn Kathir (may Allah have mercy on him) said: {Among them were those to whom Allah spoke} namely Musa and Muhammad (blessings and peace of Allah be upon them), and also Adam, as is mentioned in the Hadith narrated in Sahih Ibn Hibban from Abu Dharr (may Allah be pleased with him).
{…and He raised some of them in degree} as is proven in the Hadith of the Isra’, when the Prophet (blessings and peace of Allah be upon him) saw the Prophets in the heavens in accordance with the variations in their status before Allah, may He be Glorified and Exalted. (Tafsir Ibn Kathir, 1/670)
The Hadith of Abu Dharr (may Allah be pleased with him) to which Ibn Kathir (may Allah have mercy on him) referred is narrated in Sahih Ibn Hibban (2/76)
Shaykh Shu`ayb Al-Arna’ut said concerning it: “Its chain of narrators is very weak”. 
However, the Hadith of Abu Umamah (may Allah be pleased with him) quoted above is sufficient.
Why is Prophet Musa called Kalimullah?
Note: why Musa (peace be upon him) is called Kalimullah (the one to whom Allah spoke directly).
Shaykh `Abd Ar-Rahman Al-Mahmud (May Allah preserve him) said:
“Perhaps the reason – and Allah, may He be Glorified and Exalted, knows best – why Musa is called Kalimullah (the one to whom Allah spoke directly), even though Allah spoke directly to Muhammad and to Adam, is that Allah spoke to him directly on earth, when he was like any other human, which is in contrast to Allah’s speaking to Adam, because He spoke to him when he was in heaven, and in contrast to Allah’s speaking to Muhammad, because He spoke to him when he had been taken up, body and soul, to heaven. But when He spoke to Musa, he was on earth, and this is something that was unique to Musa (blessings and peace of Allah be upon him and upon our Prophet). (Taysir Lum`at Al-I`tiqad, p. 152)
For more details, please see the following answers: 12423 , 21524 , 10669 , 7459 , 143020 .
And Allah knows best.