What is the legitimacy of specialization in Sharia or worldly sciences, and how does Islamic law evaluate devotion to this specialization?

There is no harm in learning any permissible knowledge, whether religious or worldly, after having learned one's individual obligations (fard al-ayn). One will be rewarded for it if the intention is sincerely for Allah. Learning it may even become obligatory if it pertains to a communal obligation (fard al-kifaya).