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And Moses said unto him, Enviest thou for my sake? would God that all the Lord’s people were prophets, and that the Lord would put his spirit upon them! (Num. 11:29)

“Do you believe Joseph Smith, Jun., to be a Prophet?” Yes, and every other man who has the testimony of Jesus. For the testimony of Jesus is the spirit of prophecy.—Revelation, 19:10th verse. (Teachings of the Prophet Joseph Smith, p. 119)

It was asked me by a gentleman how I guided the people by revelation. I teach them to live so that the Spirit of revelation may make plain to them their duty day by day that they are able to guide themselves. To get this revelation it is necessary that the people live so that their spirits are as pure and clean as a piece of blank paper that lies on the desk before the inditer, ready to receive any mark the writer may make upon it. Yes, my brethren and sisters here, both men and women, have revelation, and I can say with Moses of old—“Would God that all the Lord’s people were prophets.” (Brigham Young, Discourses of Brigham Young p. 41)

"He [Brigham Young] is a prophet, I am a prophet, you are, and anybody is a prophet who has the testimony of Jesus Christ, for that is the spirit of prophecy. The elders of Israel are prophets.” (Wilford Woodruff, JD 13:165)

"I do not profess to be a Prophet. I never called myself so; but I actually believe I am, because people are all the time telling me that I am. I do not boast of that. I say that every man and woman who will live their religion, be humble, and be dictated by the Holy Ghost, the spirit of prophecy will be upon them." (Heber C. Kimball, JD 5:176)

"The brethren testify that Brother Brigham is Joseph's successor. You never heard me say so. I say I am a good hand to keep the dogs and wolves out of the flock." (Journal of Discourses, vol 8 p.69, June 3, 1860)

Near the end of his life, Susa Young Gates recorded that when asked if he had ever seen the Savior, Brigham responded that he hadn’t, and that he didn’t expect to until he died. (Note, 1885, Susa Young Gates Collection, Box 11, Folder 1, Subfolder 1, Utah State Archives, Salt Lake City.)

A person was mentioned to-day who did not believe that Brigham Young was a Prophet, Seer, and Revelator. I wish to ask every member of this whole community, if they ever heard him profess to be a Prophet, Seer, and Revelator, as Joseph Smith was? . . . Who ordained me to be First President of this Church on earth? I answer, It is the choice of this people, and that is sufficient." (Brigham Young, JD vol. 6, p. 319-320)

"I am not going to interpret dreams; for I don’t profess to be such a Prophet as were Joseph Smith and Daniel; but I am a Yankee guesser;" (JD 5:77)

March 1904 during Senator Reed Smoot hearings (after being president for 2 1/2 years)
Joseph F. Smith: Well, I can say this: That if I live as I should in the line of my duties, I am susceptible, I think, of the impressions of the spirit of the Lord upon my mind at any time, just as any good Methodist or any other good church member might be. And so far as that is concerned, I say yes; I have had impression of the Spirit upon my mind very frequently, but they are not in the sense revelations.

San Francisco Chronicle, 13 April 1997
 “Hinckley was interviewed last month in his room at the Santa Clara Marriott Hotel, just before he delivered an address to the World Forum of Silicon Valley.”
Q: You are the president, prophet, seer and revelator of the Mormon Church?
A: I am so sustained, yes. (Laughter)

Part 1: "Prophets, Seers and Revelators" 
Part 2: What are their Fruits 
Part 3: The Words of "Prophets"
Part 4: Comparison with True Prophets