The classical and motivating examples of a homotopy spectral sequence is the Bousfield-Kan spectral sequence which computes homotopy groups of a topological space/simplicial set/homotopy type realized as the totalization of a cosimplicial homotopy type. It may be regarded as the unstable analog of the Adams spectral sequence, which computes homotopy groups of certain spectra. The corresponding spectral sequence for homology groups of the totalization of a cosimplicial homotopy type is the Eilenberg-Moore spectral sequence.
The Bousfield-Kan spectral sequence was introduced and originally studied in
Aldridge Bousfield, Homotopy Spectral Sequences and Obstructions, Isr. J. Math. 66 (1989), 54-104.
An early textbook account of this work is in
Lecture notes include
Discussion of computations using effective homology includes
Discussion in homotopy type theory is in