Yes, which is of course a paradox. The fear of getting fears. =P
I fear failing in life, by my standards. This means getting a standard job that I'm not too interested in, and staying where I currently live.
I don't believe in the afterlife or reincarnation, so I guess you could say I fear dying too early as well, dying before I'm "done".
As for trivial things, like spiders, I can't think of anything special.