For me, it's spiders with wings.
Everyone seems to have some horrible combination of creatures, events, diseases, etc., that, if it were to happen, would surely push them over the edge of sanity. This is probably the stuff of SyFy Original Movie scripts, but on some deep, primal level, these fears show us that we know our mortality. Know it and probably hate it. Or at least hate its approach in mutated form.
How about mosquitoes and dysentery? Or the common cold and pancreatic cancer? Now we're moving into 12 Monkeys territory.
It's been noted time and again that the things we fear the most aren't always those most likely to happen. Terrorist attacks are far less likely to kill you than your average heart attack. Too bad we can't work up a morbid fear of hamburgers and soda pop.
Aside from our own mortality, we often fear change. We like things to be comfortable, or at least familiar, for the most part. Drastic changes can bring fears both legitimate and unfounded: the pure fear of the unknown has plagued our species since long before Hamlet realized he couldn't actually commit suicide, despite his otherwise depressed-emo lifestyle.
Yet, it is precisely the fact that life changes, that life ends, which gives it such meaning to us mere mortals. If we all lived forever and had nothing bad ever happen to us, could we appreciate, truly, the windfalls that come our way? Could we enjoy a day of sun as much if it hadn't just poured rain the week before? Could we love as fully in the moment if we had never lost a dear one? That bittersweet mix of emotion as we look into the eyes of our children, a combination of pure love, fear of inevitable hurts, and sheer joy at their existence--that is what it means to be alive.
I've experienced that, and been the richer for it. So bring on the flying mutant spiders.