IRRATIONAL FEARS: Clayton Cramer notes an interesting claim: Primate Cloning Can't Work. The linked article explains some of the science behind the assertion. Even if all the science holds true, of course, that just makes human cloning harder, not impossible.
I don't think too much about cloning. Even if a technique for cloning humans could be perfected, it would still be much more expensive than having kids the old-fashioned way, and it wouldn't be nearly as entertaining I'm sure. Except to mad scientists, I guess. I would certainly have objections if celebrities started growing headless clones of themselves to harvest organs from, but that type of thing is far enough away that I'm content to leave it for my grandkids. Worrying about cloning is like worrying about an alien invasion: they're both theoretically possible, but the science involved is so astronomical (no pun intended) that what's the point?
Another silly fear: malignant artificial intelligences that take over humanity. It makes for a decent movie, but it's not even theoretically plausible yet, and I'm not sure it ever will be. I'm getting my PhD in AI, and the state of the art is qualitatively distant from any sort of human-like intelligence.
How about Grey Goo? This is the fear the nanorobots will take apart all the atoms of the earth and use them to build more and more copies of themselves, thus reducing everything on our planet, and the planet itself, to a uniformly smooth grey goo. However, that scenario is impossible, due to the same laws of physics that protect us from alien invasion.
Other threats, such as asteroids and plagues, are becoming less and less dangerous as our technology improves. Frankly, the biggest danger left appears to be the possiblity of a nuclear holocaust initiated by some terrorist organization. Fortunately, we're making some headway in that realm as well.