Specificity of Protein Kinases in Signaling Pathways..?
Phosphorylation requires a nucleophile and hydroxyl oxygen acts like one. Serine, theronine and tyrosine get phosphorylated on the free OH group in their side chains. 

Nitrogen, in some cases also can act as a nucleophile. In case of histidine, the imidazole nitrogen is phosphorylated during bacterial chemotaxis signaling. As far as I know eukaryotes do not have an enzyme for histidine phosphorylation. 

Carboxylic groups in glutamate and aspartate can technically be phosphorylated as it happens in case of phosphoglycerate. Glutamate is phosphorylatyed by the enzyme glutamine synthase during the synthesis of glutamine. Same happens in the case of aspartate during aspargine synthesis. However, this is a reactive intermediate and the phosphorylation is not stable like that of hydroxyls.

Other amino acids do not have such nucleophilic groups to render an attack on the phosphorous atom in phosphate- so they are not phosphorylated.