The glycine side chain (-H) is unreactive. Alcohol side chains can be made reactive after oxidation with periodate. See this section in the Molecular Probes Handbook:

http://www.lifetechnologies.com/us/en/home/references/molecular-probes-the-handbook/reagents-for-modifying-groups-other-than-thiols-or-amines/reagents-for-modifying-alcohols.html#head1

I doubt this technique would be useful in vivo, however.

Serine can also be phosphorylated by site-specific kinases in vivo, of course.