In my lifetime sometimes I've heard stories of miraculous healings -- but nothing on the level that would be considered proof or demonstration of the supernatural. I hear rumors of things far away -- but for things where we can easily imagine a natural healing, or the witnesses are not brought forward, or the person healed isn't brought forward, or the original illness couldn't be established. Even going back into history, the evidence for miracles is generally weak -- with one notable exception.
There are a series of miracles attributed to Jesus of Nazareth, reported by his contemporaries in detail. Many of Jesus' miracles took place in crowds, where his followers may not have known the people who were healed. But there are others where the disciples passed down more information. Bartimaeus in Jericho was blind and had his sight restored. Lazarus in Bethany had died and was buried, and was raised from the dead. In Capernaum, the daughter of Jairus the synagogue ruler was raised from the dead. Peter's own mother-in-law was healed.
The early Christian church continued to look at these known miracles as showing Jesus' uniqueness. Quadratus, an early Christian, wrote:
Our Savior’s works were always there to see, for they were true – the people who had been cured and those raised from the dead, who had not merely been seen at the moment when they were cured or raised, but were always there to see, not only when the Savior was among us, but for a long time after his departure; in fact some of them survived right up to my own time. (The quote from Quadratus is preserved in Eusebius’ History 4.3).That's the kind of thing that you look for in order to believe a miracle.
An interesting thing occurred when Jesus' enemies tried to discredit him: they did not try to deny that miracles had occurred. An ancient Jewish writing, the Talmud, speaks of Jesus as a sorcerer. They tried to claim that Jesus' miracles were acts of evil. It would have been much simpler for them to deny the miracles ever happened, if they could.
Even today, in cultures that are more open-minded about what is beyond this physical world, Christians can boldly point to miracles as a way to show that Jesus is unique. You can read the entire Koran without hearing of Mohammed healing anyone, or raising the dead. Likewise Confucius and Lao Tzu and even King Solomon are not exactly known for their miracles. As mentioned before, that's not to say anything against Confucius or King Solomon; just that "one greater than Solomon is here."
But for our culture, we might do better to explain it to a skeptic the other way around: Do you assume there is nothing beyond this world because you have not heard good evidence of a miracle? Jesus' life gives us the world's most serious evidence of miracles. There are details of the conversations that took place, records of the confrontations about whether certain miracles were actually "evil". These things really happened, according to more than just Jesus' followers; also according to those who thought Jesus was out of line to do some of these things. The records of Jesus' life have to be read with an open mind. It is begging the question to bring in pre-conceptions that there is nothing beyond this physical world, before seeing the best evidence for yourself.
But shouldn't our skeptic be able to see evidence of the supernatural in all kinds of places, not just in Jesus? Not necessarily; there is another possibility: that Jesus really is unique.