Ok, I’m gonna go right out and say it: For the majority of digital photography, a UV filter is pointless. The original intent of these filters was to correct the color cast produced by hazy conditions when shooting film. With digital, you can correct color extremely easily in post by fine-tuning white balance settings or adjusting tint/hue.
The other reason we are encouraged to put a UV filter on is for “protection.” If you’re talking about impact protection, a really thin piece of glass isn’t going to do much other than break if there’s an impact. The truth is, most lenses have a robust front element that won’t easily break. If you do drop a lens with a filter attached, chances are you’ll dent the filter ring and have a very hard time removing the broken filter. Sure, there are lots of anecdotal stories out there about how a UV filter “protected” their lens from an impact, but have you ever noticed that nobody does the control experiment? In other words, what would the damage have been if the filter wasn’t there? Most of the time, you can protect your lens from impact damage by using the lens hood and a lens cap.
There are, of course, times when you do want to protect your lens from sand, salt water spray, or other elements. In those situations, a “protective” filter is perfectly reasonable. But with any filter, always make sure you get one that has anti-reflective coatings and good glass. Those $10 filters are cheap for a reason.