Darboux's theorem places heavy restrictions on the derivative of a function. In particular, it must satisfy the intermediate value theorem. A standard example of a differentiable but not continuously differentiable function is x^2 sin(1/x) (and 0 at x=0). Using this, a bump function, and a number of theorems concerning uniform convergence and differentiability, I can construct a function which is differentiable but discontinuous at points 1/n for integers n.
Can a derivative be discontinuous on a dense set? On a set of measure > 0? Are there any standard examples of particularly awful behavior in this vein?
This was a popular sci.math topic back in the days when sci.math was one of the only places on the Internet for discussion of things like this.
The answers to your questions are yes, yes, and not as far as I know--- but only for the reason that pathologies like these aren't commonly discussed enough to have "standard examples" for them. Examples do exist however and references are given in the first link below, which is a kind of survey of this topic.
To summarize the general result, a subset E of the real numbers R is the set of discontinuities of the derivative of an everywhere differentiable function from R to R if, and only if, E is meagre (that is, a countable union of nowhere dense sets) and F_sigma (that is, a countable union of closed sets). So, for example, because the set Q of rational numbers has both of these properties, there is a differentiable function f from R to R with the property that {x in R: f' is not continuous at x} = Q. It is nontrivial to construct examples of sets with these properties that have positive measure, but they do exist.
[Depending on how your construction for the set E = {1/n: n a nonzero integer} goes, it may be possible to modify it only slightly to produce an example for E = Q. See if you can generalize your construction to produce, for any sequence (x_n), a differentiable function f whose discontinuities of f' occur precisely at the points x_n and nowhere else, and then use the fact that Q is denumerable.]
Meagre sets are sometimes called sets "of the first category" (this older terminology is due to Baire who had a number of categories he placed sets in and proved theorems about). The notion of Baire category (and the "Baire category theorem") is a pretty standard tool in analysis and often comes up in the construction of ugly counterexamples to things that feel like they "ought" to be true but aren't.
The notion of an F-sigma set is also part of a more general set of notions, the so called "Borel hierarchy" of sets you can form from the open sets in a topological space by iterating the operations of countable unions and intersections. Set theorists tend to pay more attention to this (in the field of "descriptive set theory") than analysts do because fine study of the Borel hierarchy or structure of Borel sets in general is not necessary to solve the problems that analysts want to solve.