For the curious and those who seek with an open mind, I'll talk about the false prophecy that I find most striking, mostly because it can be proven using only the standard works in the LDS canon.
Doctrine and Covenants 84:1-5
1: Establishes that this is a revelation to Joseph Smith and six other elders.
2: Establishes that this is the word of the Lord concerning the last days, the gathering of the saints, and the city of New Jerusalem.
3: Establishes that the "city shall be built" at the temple lot in Far West.
4: Establishes that "the city New Jerusalem shall be built by the gathering of the saints, beginning at this place."
5: Establishes that the temple will be built there within that generation.
Does the prophecy refer to a different generation than the one living then? No. Doctrine and Covenents 115:7-12 makes it clear that the Lord wanted it built by the people alive at that time, as it specifies Far West as a holy and consecrated land, commands them (the people alive) to build a house (temple) for him, and says to "let the beginning be made on the fourth day of July next," and "from that time forth labor diligently until it shall be finished." This is directed towards the people alive then, even specifying a date.
Could the persecution that the early Mormons experienced thwart/change God's plans? No. A Mormon should remember 1 Nephi 3:7. All people should consider the long history of God's peoples triumphing through persecution, not leaving a prophecy unfulfilled, often despite great odds.
And the kicker? Revelations 21 renders the prophecy false in comparison to Biblical prophecy. The first verse establishes that there is a new heaven and new earth, for the old ones had passed away, and there will be no more sea--e.g., no continents. Its rather difficult for something to exist in a place that no longer exists, such as Missouri, or even America.
And then verse two says that New Jerusalem will descend from heaven, already prepared, rather than be built here on earth.
Verse twenty-two is probably the most telling. "And I saw no temple therein: for the Lord God Almighty and the Lamb are the temple of it." So why would the Lord ask for a temple to be built for the New Jerusalem, when there will be no temple? He wouldn't. He doesn't need to.
No temple exists in Far West as of today, long after the generation of the 1830s passed away.
I've heard many Mormons try to justify this by saying that Joseph Smith was just a fallible man and made mistakes. God doesn't make mistakes, however, and won't let people be misguided in his name. A thorough search of the references about false prophets in the topical guide will show a member that God has made that clear.
Joseph Smith, based on all this, was a false prophet.