To most pet owners, Border Collies tend to be a "love 'em or hate 'em" breed. Most people who like working dogs can appreciate the BC even if they wouldn't own one. They are very quirky though, and I've found quite a few of their traits that annoy people.
-Their energy and enthusiasm. They put 100% into everything they do (this is great when you're doing agility or herding, but not so lovely when they decide they want to scratch at the door to get in)
-Their speed. BCs don't walk, they jog or sprint. This often applies indoors when they are young, so make sure to nail everything to the floor.
-Their intensity. They are intimidating on occasion. People who don't like having staring contests or being watched will most likely be annoyed by a Border. If they only like throwing a tennis ball once, or playing tug for a few seconds, a BC will annoy them.
-Their sense of humor -- it's twisted! My pup absolutely ADORES startling one friend of mine, who thankfully is a dog person and gets a kick out of it (he actually encourages it more than I'd like, but so far Dakota only does it to him).
If both of them are on the floor (they usually are because my friend is in front of my TV quite often) Dakota will sneak around Dmitry and sit there silently until Dmitry relaxes and puts the dog out of his mind. As soon as he resumes doing whatever he was up to, Dakota pounces on him and puts an ice-cold nose right in Dmitry's ear. He does the funniest little butt-wiggle before he jumps him, and after he's successfully startled him and slimed in his ear, he backs off and just positively grins. I swear he's laughing.
Anywho.. Just about anything about the BC could be disliked by some. They're perfect for me but they're fast, hairy, mischevious, manipulating little monsters and not everyone can appreciate them.
Papillon: Most people love Ripley. The only thing that some people aren't wild about is that he's not especially friendly with strangers, but he warms up quickly. I don't have a ton of experience with the breed outside of my own dog, but my biggest issue with the Ripster is that despite my efforts, it's been 2 years and he's still not reliably house trained.