I agree that fixed gives you less flexibility to deal with an emergency. My big problem is a particularly nasty terrier at a local farm. Often, when I ride past on the bridlepath (part of an official long distance cycle and walking route) the dog runs out and tries to bite my foot. I love dogs, but this one is evil. I have tried getting off and squatting down to befriend it but it is just too aggressive. Now I try to ride through before it notices me, but once or twice it has successfully ambushed me. At high speed on fixed, taking your foot off the pedal either to get it away from the dog's teeth or, in desperation, to kick the dog away, can lead to some hilarious out of the saddle and nearly over the bars moments.

I ride with front and rear brakes on my fixed so I think that in the event of a sudden puncture, I'd not be substantially worse off than on my geared bike.