I almost hit a dog on a MUP here in SoCal last summer. Stupid owners were just walking along, clueless and the dog walked right across the path in front of me as I was coming up on them. Fortunately, I saw that the dog wasn't leashed and slowed down. If I hadn't, I would have probably killed or seriously injured the dog and gone over the handlebars. As it was I was able to stop in time. The owners apologized but that doesn't mean much IMO.
I also encountered a guy once walking a huge and extremely aggressive dog on that same MUP once. Thank god he had that stupid mutt leashed. I don't get why people keep dangerous pets...I really don't.
Personally, I think MUPs, parking lots and residential areas are probably the most dangerous places you could ride a bike. I always have my guard up when riding on the MUP and through residential areas. I NEVER ride through parking lots.