My friend and I are attempting to learn about spectral sequences at the moment, and we've noticed a common theme in books about spectral sequences: no one seems to like talking about differentials.
While there are a few notable examples of this (for example, the transgression), it seems that by and large one is supposed to use the spectral sequence like one uses a long exact sequence of a pair- hope that you don't have to think too much about what that boundary map does.
So, after looking at some of the classical applications of the Serre spectral sequence in cohomology, we decided to open up the black box, and work through the construction of the spectral sequence associated to a filtration. And now that we've done that, and seen the definition of the differential given there… we want some examples.
To be more specific, we were looking for an example of a filtration of a complex that is both nontrivial (i.e. its spectral sequence doesn't collapse at the $E^2$ page or anything silly like that) but still computable (i.e. we can actually, with enough patience, write down what all the differentials are on all the pages).
Notice that this is different than the question here: Simple examples for the use of spectral sequences, though quite similar. We are looking for things that don't collapse, but specifically for the purpose of explicit computation (none of the answers there admit explicit computation of differentials except in trivial cases, I think).
For the moment I'm going to leave this not community wikified, since I think the request for an answer is specific and non-subjective enough that a person who gives a good answer deserves higher reputation for it. If anyone with the power to thinks otherwise, then feel free to hit it with the hammer.
Best Answer
Two simple examples with lots of interesting differentials are given by the Serre spectral sequences for integer homology (rather than cohomology) for the fibrations $$K({\mathbb Z}_2,1) \to K({\mathbb Z}_4,1)\to K({\mathbb Z}_2,1)$$ and $$K({\mathbb Z}_2,1) \to K({\mathbb Z},2) \to K({\mathbb Z},2)$$ where in the second case the map $K({\mathbb Z},2) \to K({\mathbb Z},2)$ induces multiplication by $2$ on $\pi_2$. In both cases one knows the homology of all three spaces and this allows one to work out what all the differentials must be. The differentials give a real shoot-out, with nontrivial differentials on more than one page, and in the second case there are nontrivial differentials on infinitely many pages. The best thing is to work everything out oneself, but if you want to check your answers these two examples are worked out as Examples 1.6 and 1.11 in Chapter 1 of my spectral sequence notes, available on my webpage.
These examples may not really be the sort of thing you're looking for since they involve computing differentials purely formally, not by actually digging into the construction of the spectral sequence. But of course a lot of spectral sequence calculations have to be formal if one is to have any chance of succeeding.