Do you use screen protectors on your handhelds devices?
Personally, for the most part, no. My iPhone 5S never had a screen protector, and after having it for almost 2 years now (due for an upgrade in spring) still not a single visible scratch on the thing. I do imagine if I held it up to bright light I could probably find a hairline scratch or nick here and there, but nothing that I see in every day use. I do keep it in a good case and that case is still going strong after a few drops, though there are damaged spots (that's kind of the point, though).
Same goes for my iPad - no scratches or nicks I see in everyday use, though there is one little nick I can see in bright light. Granted, somedays I can't find the thing even though I know roughly where it is.
My Vita is also screen protector-less, though I keep it in a pretty heavy duty case when not in use. Again, no visible scratches. I take care of my electronics as much as I can.
The one exception is, as of about an hour ago, my 3DS. I had tried a couple screen protectors on it before, but could never get the darn things on without getting bad air bubbles, dust, or cat hair under them. So I gave up for a while. But then fairly recently, I did notice a couple of hairline scratches on the touchscreen, particularly a cluster in the upper left corner, likely from using the Pokemon-Amie features in Alpha Sapphire. I did notice early on the touchscreen seems to be some sort of plastic, so it scratched easier. Oh well - I don't have a special edition or anything, but I figured I'd try one more time to at least prevent further damage. It seemed the online consensus was HORI brand, and they released a set with an interesting tab system for applying them, which is supposed to minimize dust and air bubbles and the usual annoyances. Skeptical, but willing to try it after reading generally favorable Amazon reviews. I ordered them, they arrived today, and I applied them successfully. Their tab system, though hard to get at first glance, worked. Minimal dust, and what little got under them I could gradually push out with my debit card without doing any damage. They seem to adhere with some sort of static effect instead of adhesive. No air bubbles, totally clear, touchscreen still responsive and 3D effect (though I seldom use it) works as before. Also it makes the small scratches that were there totally invisible.