For what it's worth...
The HPV vaccine (Gardasil) has been administered to girls in all Ontario public schools for the past 10 years. Starting last year, it was administered to boys, as well.
I have never met anyone who has suffered a serious reaction to it. I've never even heard rumours of such a thing happening to any of my children's schoolmates, over the past decade. I have heard of many mild reactions (ie, sore arms, fevers), however.
I'm not saying it hasn't happened, just that based on personal experience vaccinating your kids certainly does not seem to be a risky thing to do.
Conversely, I do know several people with HPV, including one very dear friend who graphically described getting the warts burned off her genitals (eek!).
By the numbers:
In 2015, approximately 9 million publicly funded doses of vaccine were distributed in Ontario (this number includes, but it not limited to, Gardasil).
231 sore arms were reported (to be fair, this number is probably vastly under estimated, as most people won't bother reporting a sore arm)
167 rashes (more likely to be reported, as any doctor the child sees is required to report it)
117 skin reactions (ditto)
34 "serious events" were reported (ie, anaphylaxis), representing 4 in every 1 million doses distributed.
There were no deaths attributed to vaccines in Ontario that year.
Beyond that, though, I know I am not at all qualified to research the safety of a vaccine. I get my daughter (3rd year Biochemistry) to look up most of my science related questions for me, and even she admits she doesn't have the background to fully understand the research around this. And she knows so, SO very much more than I ever will. I think, sometimes, the egalitarian nature of information online leads us into thinking we can understand everything if we just do a little Googling. I've been mislead more times than I can count by something I've read that sounds plausible. Which is why I am willing to trust the expertise of my public health officials, here in Canada. They've given me no reason not to.
Heck, I even get my flu shot every year, now.