Gotta agree w/ Jeanie that burying your ends is the best solution. Of course, part of this problem depends on what gauge wire you're using. Can you fill us in on that?
After I finish a section I generally make a triple wrap & then poke the end into another loopy section or bead. I make sure that the end is away from the skin & as short as possible. If I'm feeling insecure about the wire holding, I may do the triple wrap twice a short distance apart.
Another little inexpensive tool has a burr cup on the end that will round off the edges of the wire. I've only found it in one size, so it does matter what gauge you're using [larger vs smaller]. The burr cup does work on a number of smaller gauges. It can be easier to handle than the files if the wire is narrow & hard to get at or colored.
sher