The Supernatural is too broad of a spectrum. It's common for many people to group everything that is unexplained or paranormal together, but, I don't think that should be the case.
Therefore, there are parts of 'supernatural' I think do, or could exist, while others, I don't or don't think they could.
I'll break it down here:
Monsters: Maybe, maybe not. Creatures like Loch Ness or Bigfoot could exist, but that doesn't mean they do, mostly due to not having valid credibility in photos or eyewitness sightings. (BTW, that picture is a tarp. I keep telling people it doesn't even look organic, but people will believe what they will.
Paranormal: Even this 'subcategory' is way too broad to get a definite answer to me.
There's a difference between a Ghost, a Demon, and a Lightning Orb, but people seem to group them together anyway.
'Ghosts': Yes. I've had way too many experiences with Spectral Anomalies (as they SHOULD be called) to not believe.
Demons/Angels/the Underworld/Heaven: These I fully believe in, because I'm Christian. But, I believed in demons and angels even before I was a Christian. Why? I don't know you well enough to get into that :p
'Lightning Orbs', 'Rods', 'That-thing-you-see-out-of-the-corner-of-your-eyes-but-is-gone-when-you-look': Also known as the Metaphysical. Yes, in a sense. Scientifically, It's possible for Lightning or Plasma Orbs to manifest, and there is an abundance of evidence to prove it. Rods and the Other Thing, I'm more skeptical about. In this case, It's been proven to be poor photography and vision errors.
Psychic Capabilities: In a word, yes. Sure, it's a little out-of-the-ordinary, but why can't it exist?
Horoscopes and Astrology, Seances, Ouija Boards, Crystal balls : IMO, pure bunny beans. All these stem from the desire to believe, and little more.
Magic: Depends. A lot of magic happens to be proven science disguised as magic. Some things might exist. Some might not. It's really hard to research something like that.
Aliens: I wonder if ET needed a galactic visa to enter the solar system. Ok, bad joke. I only believe that they could exist. Not that they do exist.
Government Conspiracies: I think it's absolutely hilarious when someone can look at cold hard facts, find the tiniest thing wrong, and find a way to blow it out of proportion. Sure, the government doesn't tell us everything (see 'Aliens'), no government does. But the lack of evidence does not disprove. And some of the time, over-abundance of evidence disproves itself.
In closing: Don't hold your ground if your faced against an Atomic Warhead, just because you don't think it exists. You could end up vaporised by nothing. (Get it? Oh the wit.)