The fox is one of my favorite parts too. It is almost like Tolkien decided one day, 'Hey, I'm just going to stick a fox in here.'
Another part that I like is just after Frodo wakes up after Gildor had left and Pippin is trying to talk to him and Frodo says that he just wants to think while he is eating.
The part about Bilbo giving out gifts to people when he left the Shire is also a part I always enjoy, especially the part about Lobelia's spoons.
I can't think of anymore right now. I know there are more but as they are the "little and unimportant parts", no more are coming to mind.