Question: It is very tempting to assign extra credit problems that are (a) substantially more difficult than others on the problem set, (b) ask for the history of something we discussed in class, or (c) are related to something just barely outside of the scope of the class.

However, I have never done so because, in my experience, very few professors give extra credit.

* **Why do so few professors assign extra credit (or is this not the case)?**
* **What are the arguments against assigning extra credit problems?**

Answer: 
I dislike "extra credit" for many reasons, many of which have been mentioned already. But mostly, I think that the argument of *Students like bonus problems...I think they [with the reward of extra credit] encourage doing more than was necessary while studying* (which I believe to be roughly the standard argument, but this is (essentially) a quote from Tim Seguine in the comments) just isn't good enough. In the real world money is not the only motivation, and I would not like it if, for example, the police force was driven by money rather than a desire to make where I live a safe place. Similarly, if we replace "money" with "final grade" then I do not think it is healthy to motivate students solely by their final grade. I believe an alternative motivation can and should be used in place of "extra credit". For example: Chocolate.

More concretely, I tell my students that if they can find an error in my notes or in my problem sheets then they should come to my office and I will reward them with chocolate (in their answer, vonbrand says that they reward this with extra credit). This means that they can argue all they like about whether or not they deserve chocolate (they point out the errors, but they do it in lectures!), but ultimately the arguments are in good humour as they are arguing over...a piece of chocolate. Additional "extra credit" questions which are meant to motivate the students now become competitions where the winners get, say, a not-quite-the-cheapest box of chocolates. Now, because the rewards are chocolate everything is fun while the students remain motivated (well, enough of them remain motivated to keep me happy), so this all adds to the student experience. I believe that if I replaced "chocolate" with "extra credit" then there would be no improvement in the student's grasping of the subject and I would enjoy lecturing *much* less.
I give extra credits routinely, mostly to reward not-asked-for work (point out an error in the class notes, suggest an interesting alternative solution for an exam or homework problem, ...) and sometimes for extra homework assigned. In my (by now extensive) experience, students who go the mile to get extra credit would get good or outstanding grades even without the extra points. A very few of those hanging in the balance get over the bump by this, but then they have shown enough insight to merit a passing grade.

In exams I asign a few extra points, but not a problem marked specially. That gives some leeway for mistakes and still getting a perfect score. My exams are typically a fairly simple, general question ("do you even know what we are talking about?") and the remainder several more difficult problems covering the material about evenly. Sometimes questions have a "bonus track" part, if it makes sense to look somewhat further.