I wanted to compile a list of key points that, for me, make the Mormon church absolutely undeniably false. This will make for an easy thing to share with those that call into question my change in beliefs, as well as a few good talking points with those that are starting to question their belief. So without further ado, here is the list.
1. Joseph Smith's Polygamy/Polyandry
Among members of the LDS church, Joseph Smith had a large number of polygamous adventures. He had over 30 wives (something like 33). Some of these wives were as young as 14, and some were still married to their (living) husbands. Some of the women Joseph Smith married after he had sent their husbands out on missions. Does this sound like the behavior of a prophet? Definitely not, especially when he claimed not to have practiced polygamy, and the church still claims (for the most part) that he was either not married to more than one wife, or that he was celibate with those plural wives (which is demonstrably false).
More info: www.wivesofjosephsmith.com
2. The Book of Abraham is a Proven Fraud
The original papyri that Joseph Smith used to translate the Book of Abraham have been found, quite a while ago, actually. Due to the finding of the Rosetta stone and the ability to translate Egyptian, the Book of Abraham has been translated into English by less than divine means. It turns out that the Book of Abraham was nothing more than a common funeral document buried with mummies, that the facsimiles were incorrectly finished (they were found damaged and incomplete, but there are other copies of the same thing) and incorrectly translated (just one example, the "high priest" performing the sacrifice was supposed to be Anubis). Also, the funeral document was written long after the time of Abraham. Finally, the Egyptian alphabet that Joseph Smith was creating was completely off from the actual translation. Three strikes, and you're out.
The kinderhook plates are a corollary to this. Look them up. Those and the Book of Abraham together show that Joseph Smith couldn't translate a thing, whether with divine assistance or not.
More info: http://www.bookofabraham.com/
3. Exponential Decay of Prophecies
I have touched on this previously, so I will keep this one brief. Why is it that Joseph Smith had tons of prophecies, many of the false ones having been covered up, and every single one of his successors had progressively fewer prophecies. In fact, the number of prophecies follows a power law curve (in case you cared). Why is it that the most significant prophecy in the past 30 years was either the lowering of the missionary age (can you even call a policy change a revelation?) or the declaration to the world on the family (the family is important, nothing new here). Prior to that, the only revelations since Joseph Smith have been squashing polygamy (more likely due to social pressure, the prophet was about to be arrested after all), and allowing blacks to have the priesthood (more on that in a sec).
4. Rampant Racism and Sexism
The church is filled with a history of racism and sexism, that still lasts until today. Church leaders used to teach that blacks were black because they were less valiant in the premortal existence, were descendants from Cain (this comes from the Book of Abraham) and thus weren't entitled to hold the priesthood, and that they would never hold the priesthood. This really speaks for itself rather well, anyone that resolves this uses a lot of mental gymnastics in order to overcome this sticky point. In addition to this, the church treats women as second class citizens. They are only allowed to hold positions of authority over other women or children. Their only place (it is taught that this is the best thing they can do with their life) is to be a mother and to raise their children. They are essentially discouraged from having a career, and are looked down on if they don't get married and have children. They are seen as objects to be passed around between men (if you are faithful, you will be given to a faithful priesthood holder in the next life). In the temple women make covenants to obey their husbands and not God, we also learn in the temple that women will be allowed entrance to heaven by their husbands (when getting married, the husband stands in the place of God for his wife at the veil).
5. Book of Mormon Anachronisms and DNA
The Book of Mormon has several problems. First, DNA analysis on Native Americans shows that they have Asian roots and not Jewish roots. Look this up, there are several angles to tackle this, and none of them work in favor of the Book of Mormon. Second, there are a ton of anachronisms in the Book of Mormon, meaning there are things that the Book of Mormon references that Joseph Smith, while writing it, would have known about, but that didn't exist in the time or place of the Book of Mormon. There are references to animals and crops, and there has been no evidence for those animals (such as horses) or those specific crops being available at that time in the Book of Mormon. There are references to materials (e.g. steel) with no evidence that the steel was there, no evidence of the facilities existing to make the steel, no evidence of the techniques to make steel, and no evidence of the waste created when making steel. There is more, but studying this out has not been a strong point of mine. In addition to this, there is no evidence of the many battles that took place. There were supposedly millions of people slain near New York (there are many other locations), but not one shred of evidence of there being millions of bodies, or other evidence of remains being there. This is not the only large scale battlefield. How do the traces of two civilizations of that supposed size simply disappear without a trace? Also, the population growth necessary to reach the numbers of people that died in these battles is simply unheard of, especially with the numbers of deaths found in the battle portions of the Book of Mormon.