What is the best evidence that UFOs exist? The best evidence that UFOs exist is what the letters stand for: Unidentified Flying Objects! It is simple, if it is flying and can't be identified, it is an authentic UFO. Many people see UFO's all the time and, as any court can tell you, witnesses are evidence. What would be the best evidence that UFOs which have come from other planets exist and have visited the Earth in the past, or are visiting it in the present? That is a different question all together! The answer to that question depends upon what you want to believe!
But this is about evidence rather than belief, so how can the beliefs affect evidence? Evidence means "that which is seen"; it doesn't mean "that which is thought about and speculated on". It should be scientifically valid. One problem is that people often see what they want to see, and they want to see what they believe, in this case Alien Spaceships!
The use of Mathematics is a good example, while they sound factual, no one really knows anything about other planets or the chances of civilizations arising even if there was life out there; with UFOs the math is currently all speculation! Based upon mathematics and current astronomy trends at finding other planets there is likely to be life out there, other civilizations may have developed. But, as any good scientist will tell you, speculation is theories and not evidence. Everybody likes to speculate or postulate what may be, but evidence is about what exists, mathematics is about the chance of it existing.
So people can create their own evidence! Let's look at some of the records from the past and decide if they have evidence or speculation. Why do this? In the ancient past, no one knew about life on other planets, so where could the records have come from? If UFOs have been here in the past they could be here now.
Zecharia Sitchin's Book, the Twelfth Planet, has sketches from a great many ancient texts showing what Sitchin believes are Flying Saucers and people in what could very well be space suits. There are quite a few things that do look like space ships and astronauts, as well as mention of people flying. These manuscripts are thousands of years old, coming out of Babylon Tablets, Egyptian Pyramids, Hebrew Scrolls and several other documents. Sitchin believes that they are proof conclusive that UFOs exist and have been visiting Earth for thousands of years. Other scientists agree that the writings exist and are very ancient, but believe these hieroglyphics merely portray man's desire to fly like the birds or be raised above his fellow man. They feel it is wishes rather than fact and also say the other symbols are being mis-read. Unfortunately, either one could be right, or possibly even both! It is odd that so many random cultures would create the same ideas, but this may be an example of a basic human desire. So, does this count as the best evidence?
Then there are the ancient legends from the Dogon Tribe of Mali in Africa. These legends, as recorded by French Missionaries in 1828, claim the Dogon were visited by "Fish men from the Stars" (not Heaven, an important distinction). According to the Dogon, these visitors rode a great ball of fire to the earth that landed in the water and turned to stone. From the stone came forth fish men that talked with them and told them about the stars. These travelers gave them pottery and charts that still exist today, and have been dated at over 5000 years old! The pottery is high glaze enamel, where did the Dogon get it from? The charts are of star systems we are just learning about today. How did a primitive tribe of Africans with a life expectancy of less than 35 years know about a White Dwarf Star that circles Sirius every 50 years (The Hubble telescope put it at 50.04+/-.09 years)? The charts and pottery are physical, easily dated and about something which we are just discovering. For many people, this is the best evidence for past visitations!
What is the best evidence for UFO existence in present times? That is hard to determine. Maybe radar imagery from airports or videos, but some say those can all be faked to generate publicity for the city, so may not be "real" evidence. Would the best evidence be the thousands of sightings from reputable sources such as teachers, trained pilots and policemen? These people put their jobs and reputations on the line to make their reports. Why should they lie? Could it be the "alien abductions"? Maybe it would be the Roswell, New Mexico, records, if they are ever produced. "Angel Hair" falling from the ships has been cited as evidence, but it doesn't seem to hang around long enough to be shown or analyzed!
I am afraid the best evidence, and the only one acceptable to most people, would be a UFO landing at a major city and putting up signs "We're here". It would have to be demonstrably different from anything known and have technology never seen on Earth. Everybody would need to see it themselves. Though, even then many would claim that was just filming for the next Star Trek movie! Evidence is what one sees for themselves and is willing to accept as the truth.