This might be is more than you bargained for, but it's too good an opportunity to pass up plugging a couple cool and readable papers with impact beyond containing computable spectral sequences. In the 1980s Ravenel and Wilson famously used Hopf rings to compute some extraordinary homologies of a variety of families of infinite loopspaces. When In the target family of specific case where the loopspaces are Eilenberg-Mac Lane spaces, they used a bar spectral sequence (which arises as a filtration spectral sequence), together with Hopf ring information, to compute $E_* K(G, *)$ for various $E$ and $G$. One of the cool features of their work is that everything involved is explicit and identifiable.^{‡† You might try:}

Doug Ravenel and Steve Wilson, The Morava $K$-theories of Eilenberg-Mac Lane spaces, published in 1980 in the American Journal of Mathematics, vol. 102, no. 4, pages 691-748

for a mildly complicated but very rewarding example. Or, the algebras $H_*(K(\mathbb{Z}/p, *); \mathbb{Z}/p)$ (along with a million other things!) are computed in Wilson's very exceptionally nice book

Steve Wilson, Brown-Peterson Homology: An Introduction and Sampler, published in 1980, no. 48 in the CBMS series of conference notes,

which uses all the same machinery as the Morava $K$-theory paper but uses employs singular homology and tells you about some algebras which you already understand. These The familiarity of these two things will probably ease digestion of the ideas.

^‡

^† -- Well, this is kind of a lie, since they argue the existence or nonexistence of some of their differentials are argued by knowing what the $E^\infty$ page must look like together with some kind of sparseness of the $E^2$ page. It's unclear to me whether you want to qualitatively see everything that's going on or if However, the way they build the spectral sequence does actually give you want to start with expressions describing a formula for the boundary maps, apply them to everything in sightdifferentials, and see what differentials it's certainly possible, if difficult, for you get. I personally have only successfully used to make the latter approach relevant calculations once , and not in a way I'd want you read their argument so you know whereabouts to share; look. At small primes (and small heights, in my experience there's typically too much information running around to do that to any serious effectthe Morava K-theory paper), this is probably even accessible.

This might be more than you bargained for, but it's too good an opportunity to pass up plugging a couple cool and readable papers with impact beyond containing computable spectral sequences. In the 1980s Ravenel and Wilson famously used Hopf rings to compute some extraordinary homologies of a variety of families of infinite loopspaces. When the target family of loopspaces are Eilenberg-Mac Lane spaces, they used a bar spectral sequence (which arises as a filtration spectral sequence), together with Hopf ring information, to compute $E_* K(G, *)$ for various $E$ and $G$. One of the cool features of their work is that everything involved is explicit and identifiable.^‡ You might try:

Doug Ravenel and Steve Wilson, The Morava $K$-theories of Eilenberg-Mac Lane spaces, published in 1980 in the American Journal of Mathematics, vol. 102, no. 4, pages 691-748

for a mildly complicated but very rewarding example. Or, the algebras $H_*(K(\mathbb{Z}/p, *); \mathbb{Z}/p)$ (along with a million other things!) are computed in Wilson's very nice book

Steve Wilson, Brown-Peterson Homology: An Introduction and Sampler, published in 1980, no. 48 in the CBMS series of conference notes,

which uses all the same machinery as the Morava $K$-theory paper but uses singular homology and tells you about some algebras which you already understand. These will probably ease digestion.

^‡ -- Well, this is kind of a lie, since the existence or nonexistence of some of their differentials are argued by knowing what the $E^\infty$ page must look like together with some kind of sparseness of the $E^2$ page. It's unclear to me whether you want to qualitatively see everything that's going on or if you want to start with expressions describing the boundary maps, apply them to everything in sight, and see what differentials you get. I personally have only successfully used the latter approach once, and not in a way I'd want to share; in my experience there's typically too much information running around to do that to any serious effect.