My motivation for this question is twofold: On one hand, I'm studying algebraic topology, where - at least in the book written by Hatcher - there is quite a lot of handwaving (e.g. maps are continous because it's obvious from the picture). On the other, I'm going to be a TA next term, where it might come in handy to have some examples of handwaving gone wrong.
I'd like to see two types of examples:
1) Tell me about typical handwaving-related mistakes which math undergraduates are prone to.
2) Are there examples of handwaving in more advanced math that was possibly published and only the peer-review process uncovered it? The longer it took for somebody to notice, the cooler I'd consider the example.