There is really just one method: Phospho-specific antibodies, so you won't find a review. Historically people might have used 32P labelling to monitor phosphorylation, then IP-Western with a phospho-Tyr antibody was used for a while, then mutational analysis Ser-to-Ala, of a protein was used to demonstrate something was physiologically significant. Today people just use an antibody specific for the phosphorylation site if they want to see if a protein is phosphorylated. The western blot is the standard way to observe phosphorylation of a protein, as you can see whether the antibody recognized a protein of the proper molecular weight. ELISAs can also be useful as can immunostaining (IHC, ICC, IF) methods if you want to see subcellular localization or if you only have paraffin-embedded tissues to work with. There are some mass-spec based methods as well. The main challenges of using phospho-specific antibodies is that western blotting is non-quantitative and a relatively noisy assay when inexperienced users do it. The other major challenge is that the majority of antibodies do not actually work correctly. Phospho-Tyr antibodies in particular are notoriously non-specific for their target, so if you were to treat cells with sodium vanadate most phospho-Tyr antibodies will show you a smear of Tyr phosphorylated proteins.

If you study a particular signaling pathway, then you can the appropriate reporter assay. For more general approach, have a look at RPPA:http://en.wikipedia.org/wiki/Reverse_phase_protein_lysate_microarray