There are a lot conspiracy theories out there, most of which have no evidence to support the claims made, either because whatever evidence that has been put forth has been debunked, or no evidence has ever been put forth in the first place.
In fact there are some conspiracy theories that have no reason to continue to exist, or have no reason to exist in the first place, such as:
Moon Landing Hoax
Perhaps one of the older conspiracy theories out there, there are a lot of people who do not believe we went to the Moon, and that all of the videos (the hundreds of hours worth) and photos (the many thousands of them) taken from the Moon were all done on a sound stage.
The reasoning behind this is that it is believed by people who claim we did not go to the Moon that we did not have the technology to go to the Moon.
The problem with this argument is that we actually did have the technology to get to the Moon. Also, as surprising as this may sound, we actually didn't have the technology to fake going to the Moon.
There is also a ton of other evidence that says we did in fact go to the Moon, such as several tons worth of rocks and dirt that were brought back, the fact that not one of the hundreds of thousands of people who worked on the Moon landing project has ever said we didn't go to the Moon, or that the Soviets never said that we didn't get there, or the fact that the landing sites have been photographed by satellites orbiting the Moon.
9/11 conspiracy theories
Ever since that tragic day over 12 years ago there have been multiple conspiracy theories put forth concerning what happened that day, and while all of them tend to be different (from both who did it to how it was done) they all have one thing in common: They have all been debunked.
I know, a lot of people in the 9/11 "Truth" movement will say otherwise, and will claim that they have "evidence" that backs up their claims, the facts are is that when this so called evidence has been examined it's been shown to be either incorrect, or completely false, and it is now seriously considered by skeptics and debunkers that the only reason why anyone would continue to make these 9/11 conspiracy theory claims is that they are either self deluded, or mentally ill, or they are lying.
Autism - MMR vaccine connection
Ever since 1998 when Andrew Wakefield wrote and published a "research" paper in The Lancet that concluded that there was a "connection" between the MMR vaccine and autism (research of which has since proven to be both unethical and fraudulent and resulted in both the research paper being formerly retracted and Mr. Wakefield's name being removed from the General Medical Council, which is the British equivalency of having one's medical license revoked) there has been a conspiracy theory going around concerning the alleged connection and vaccine manufactures trying to suppress such information.
Besides the fact that none of this "information" has ever been suppressed, it has been proven by multiple scientific and medical research institutions that there is no connection what so ever between any vaccines and autism, and that all of the claims made by the anti-vaccination movement are wrong and false (and dangerous).
Project Blue Beam
Project Blue Beam was a conspiracy theory that was first brought to public attention in 1994 by an investigative journalist by the name of Serge Monast in which he claimed that NASA would use advance technology to simulate the second coming of Jesus Christ in order to start a New World Order.
So how much of this is true?
None of it.
Monast had no evidence what so ever of Project Blue Beam, or that anything like it even existed, and there is still no evidence to this day that there is or ever was a Project Blue Beam. Plus he also made several claims of when it was to implemented (1995, 1996, and 2000), all of which have clearly never happened.
The chemtrail conspiracy theories first showed up in 1996, but hasn't really gained popularity amongst conspiracy theorists until about the last few years.
The basics of the conspiracy theory is that the government, or Illuminati, or some other group, is spraying chemicals from airplanes, and that what we believe to be contrails coming off of those planes are actually these chemicals that are being sprayed around the world some sort of malevolent purposes (what ever those purposes may be, as no one can seem to agree upon what they are for).
This conspiracy theory, despite how obviously easy it would be to prove it was true, has never been proven to be true, and that all of the claims made about what chemtrails are suppose to do have been proven false (sometimes at a level in which it is considered to be laughable).
President Obama was not born in America
Even before President Obama took the oath of office in 2009 there has been this huge conspiracy theory floating around that the President was not born in the United States, and is not an American citizen, and that there has been this huge cover up in order to keep him in office. The people who make these claims often times use a "Kenyan" birth certificate that has his name on and date of birth on it as "proof" of this.
Of course there are a few problems with this:
First, that "Kenyan" birth certificate is actually Australian, and it is an admitted hoax.
Second, President Obama has released his long-form birth certificate. The birth certificate that was certified by Hawaii Governor Linda Lingle, a Republican.
So what evidence is there that President Obama wasn't born in the Untied States? None.
Every single claim of this conspiracy theory has been disproven, and every single time it's been taken to court, the judges have thrown it out and have even fined and imposed sanctions against some of the people who brought it to court for wasting the court's time.