What are the risks of fenders/mudguards? I've seen fender/mudguard descriptions that mention things like breakaway stays and discussions about the relative safety of metal vs plastic fenders, but I'm not getting what the concerns are.

Can anybody explain what the potential hazards of fenders are and how to mitigate them? The rainy season is coming…

I'm especially interested in full fenders on a road bike.
First, if you're using a conventional cantilever brake (i.e. with a yoke, not a link wire), you need a fender or reflector or something to avoid the yoke catching on the tire.

As usual, Sheldon is excellent. 

There are several types of fenders:

Downtube mudguards:



Seatpost fenders:


Both of these options sit away from the wheel, clipped on. They can't interfere with the wheel, so coverage isn't as good (esp. for water being thrown up by your front wheel hitting your feet). However, they can work on all bikes. They don't prevent your drive train from being covered in crud from the road (which reduces the lifetime of your parts) or water getting in your headset bearings. There aren't safety issues with these (they can fall off, but nothing should happen if they fall off; they can't jam in a wheel). 

Full Fenders: 


Ideally you want a mudflap on the front. These mount to eyelets on the frame and fork normally, and at the brake bolts (or reflectors or ziptie there). There are also a few clip on ones for this.  These prevent crud from being thrown up to your drivetrain and getting into the headset bearings.

The problem with full fenders is that something can get caught between the fender and the wheel or the fender can slip (esp. in the front) and jam a wheel. Then you crash. This is why you see fenders with safety release tabs so the fender will shear off in some way if something tries to get caught in the wheel rather than jam it and crash.