What is study of ancient inscriptions called ?

Epigraphy

The answer is Epigraphy.

Epigraphy is the study of inscriptions, or epigraphs, as writing; it is the science of identifying graphemes, clarifying their meanings, classifying their uses according to dates and cultural contexts, and drawing conclusions about the writing and the writers. Specifically excluded from epigraphy are the historical significance of an epigraph as a document and the artistic value of a literary composition. A person using the methods of epigraphy is called an epigrapher or epigraphist.

Herpetology is the study of amphibians and reptiles. Genealogy is the study of family lineages. Histology is the study of tissues.