JOEL -
(1) the "Grease Spot" prophecy (H.C. 6: p. 116)

"While discussing the petition to Congress, I prophesied, by virtue of the holy Priesthood vested in me, and in the name of the Lord Jesus Christ, that, if Congress will not hear our petition and grant us protection, they shall be broken up as a government, and God shall damn them, and there shall be nothing left of them - not even a grease spot. (History of the Church (HC), vol. 6, p. 116)

The footnote to this "prophesy" explains:
"This prediction doubtless has reference to the party in power; to the "government" considered as the administration; not to the "government" considered as the country; but the administration party, the Democratic Party, which had controlled the destiny of the country for forty years. It is matter of history that few years later the party then in power lost control of the national government, followed by the terrible conflict of the Civil War. The Party against which the above prediction was made so far lost its influence that it did not again return to power for a quarter of a century; and when it did return to power it was with such modified views as to many great questions of government, that it could scarcely be regarded as the same party except in name."(HC 6:116)

But many critics of the church will not accept this explanation, insisting that the entire government should have collapsed rather than just the Democratic party. This is a matter of opinion.

(2) The prophecy in the History of the Church volume 1 page 315-316 regarding destructions to greet Joseph Smith's generation.

"And now I am prepared to say by the authority of Jesus Christ, that not many years shall pass away before the United States shall present such a scene of bloodshed as has not a parallel in the history of our nation; pestilence, hail, famine, and earthquake will sweep the wicked of this generation from off the face of the land, to open and prepare the way for the return of the lost tribes of Israel from the north country. The people of the Lord, those who have complied with the requirements of the new covenant, have already commenced gathering together to Zion, which is in the state of Missouri; therefore I declare unto you the warning which the Lord has commanded to declare unto this generation, remembering that the eyes of my Maker are upon me, and that to him I am accountable for every word I say, wishing nothing worse to my fellow-men than their eternal salvation; therefore, "Fear God, and give glory to Him, for the hour of His judgment is come." Repent ye, repent ye, and embrace the everlasting covenant, and flee to Zion, before the overflowing scourge overtake you, for there are those now living upon the earth whose eyes shall not be closed in death until they see all these things, which I have spoken, fulfilled. Remember these things; call upon the Lord while He is near, and seek Him while He may be found, is the exhortation of your unworthy servant." (HC 1:315-316)

The "scene of bloodshed" in this prophecy made in 1833 was no doubt refering to Joseph Smith's 1832 prophecy(D&C 87) about the Civil War which began in 1861 and resulted in the bloodshed and death of about 400,000 people. Since this war happened only 28 years after the prophecy, there were probably many thousands of people from Joseph Smith's "generation" who were involved in it.
There are other scriptures relating to this in D&C 42:64 and D&C 45: 63-64, which also speak of coming wars and exortations to the Saints to gather in the west.
There were also instances of pestilence, hail, famine, and earthquakes which occured within that generation.

Major earthquakes in the US greater than 6.5 on Richter following this prophecy:
1857 - Fort Tejon, California
1872 - Owens Valley, California
1886 - Charleston, South Carolina.
1884 - New York City area
1892 - Imperial Valley, California
1895 - Charleston, Missouri
1897 - Giles County, Virginia
1906 - SanFrancisco, California

So there were people living at that time who did see all of the things predicted by Joeph Smith. But things like this don't impress the critics much(not even the Civil War prediction) for they will claim that most of these things were happening(or people knew they were going to happen) long before Joseph Smith's prophecy. And they are right to a degree, because I think mostly that in this prophecy Joseph Smith is just reminding us of the fulfillment of the already existing Biblical prophecies about the destruction of the wicked preceding the second coming of Jesus.

"And ye shall hear of wars and rumours of wars: see that ye be not troubled: for all these things must come to pass, but the end is not yet.
For nation shall rise against nation, and kingdom against kingdom: and there shall be famines, and pestilences, and earthquakes, in divers places."(Matt 24:6-7)

Critics base their claims of false prophecy against Joseph Smith on the following scripture:

"When a prophet speaketh in the name of the LORD, if the thing follow not, nor come to pass, that is the thing which the LORD hath not spoken, but the prophet hath spoken it presumptuously: thou shalt not be afraid of him."(Deut 18:22)

They claim that if any single one of Joseph Smith's prophecies do not come to pass in the way they think it should, then we should label him a false prophet. But their strict interpretation of this scripture would condem many Biblical prophets to be labeled as false prophets as well. Keeping this in mind we need to define a few things about prophecy.

1. A prophecy is a prediction of things that will happen in the future if certain conditions prevail. If man with his free agency decides to change his ways God can reverse or cancel a prophecy if it serves His greater purpose.
An example is found in the story of Jonah, who was told by God to prophecy to the people of Ninevah. Jonah prophesied that the people would be destroyed in 40 days (Jonah 3:4) - no loopholes were offered, just imminent doom. God changed things, however, when the people repented and He chose to spare them - much to the chagrin of that imperfect (yet still divinely called) prophet, Jonah.

And in Jeremiah 18:7-10:

"At what instant I shall speak concerning a nation, and concerning a kingdom, to pluck up, and to pull down, and to destroy it;
If that nation, against whom I have pronounced, turn from their evil, I will repent of the evil that I thought to do unto them.
And at what instant I shall speak concerning a nation, and concerning a kingdom, to build and to plant it; If it do evil in my sight, that it obey not my voice, then I will repent of the good, wherewith I said I would benefit them."

In the D&C:

"Wherefore I, the Lord, command and revoke, as it seemeth me good; and all this to be answered upon the heads of the rebellious, saith the Lord." (D&C 56:4)

2. Prophecies are fulfilled in the Lord's own time not when man thinks it should be fulfilled. The word "generation" is an unspecified period of time that can encompass just a few years or a thousand years (eg. an entire dispensation).
In Matthew 24:34, Mark 13:30, and Luke 21:32, Christ makes prophecies that are still not fulfilled which involved "this generation". The standard used to make a false prophet out of Joseph Smith would also reject Jesus Christ.

3. Only God knows when a prophecy has been fulfilled or not:

"Knowing this first, that no prophecy of the scripture is of any private interpretation.
For the prophecy came not in old time by the will of man: but holy men of God spake as they were moved by the Holy Ghost.(2 Pet. 1:20-21)

4. The text of a prophecy can undergo alteration to the point that it no longer reflects the original intent of the prophecy. This could be the case with several of Joseph Smith's prophecies and even Biblical prophecies that were either incorrectly recorded or translated.

5. According to Joseph Smith, "A prophet is only a prophet when he is acting as such" (Teachings of the Prophet Joseph Smith, p. 278).

"When the, the Spirit of truth, is come, he will guide you into all truth; and he will shew you things to come" (John 16:13).

Prophets are human and can make mistakes. The Holy Spirit may not always be with them, and it is at moments like these when prophets like Joseph Smith might predict something that doesn't happen.
Fortunately we now have an excellent system of a First Presidency and Council of Twelve Apostles who can receive the same revelation that the current President gets, and validate that it is a true revelation or prophecy.

Of course this last one won't impress any critics. In my opinion there are only a few of Joseph Smith's prophecies that may be questionable at best, because of the above reasons, but don't fall under the category of "false prophecy".

NEPHI - About the "Grease Spot" prophecy:
Joseph made a petition to the GENERAL GOVERNMENT, not to the ruling Democratic party, thus isn't the note in the history of the Church is inaccurate as the whole Congress had to have been collapsed, not just the ruling Democratic Party? Ditto for the prophecy, under this light?

JOEL - Like I said, this is a matter of opinion as to whether the demise of the Democratic party constitutes fulfillment of the prophecy.
"Knowing this first, that no prophecy of the scripture is of any private interpretation."(2 Pet. 1:20-21)

Once again the scripture critics use as their test for a false prophet says:
"When a prophet speaketh in the name of the LORD, if the thing follow not, nor come to pass, that is the thing which the LORD hath not spoken, but the prophet hath spoken it presumptuously: thou shalt not be afraid of him."(Deut 18:22)

Their interpretation claims that if any prophecy does not come true, that makes the person a false prophet. But this is not consistant with what this scripture or the rest of the Bible says. The scripture still calls the person a prophet. It does not say that he is a false prophet; it only says that if he "hath spoken "it" presumptuously" then we should pay no attention to him, regarding that particular prophecy(it). The failed prophecy is false, not the prophet.
The attitude that a prophet must be completely infallible and never make mistakes is not consistant with the rest of the Bible. Jesus was the only perfect person to live on this earth and some of His prophecies that were supposed to happen in "this generation" have not yet occured(Matt 24:34, Mark 13:30, Luke 21:32).

If you read a few entries from Joseph Smith preceding this prophecy(HC vol 6. pp. 111-115) you will see how upset he was at the government for not helping them and that he was also extremely sick the day before. In my opinion it is possible that Joseph Smith was so upset that he may have not exactly been acting as a prophet at the moment he said this, but rather it was an emotional response from a leader fearful about what might happen to his people if they received no help. I am willing to cut him some slack on this one.