Hi, I am wondering if anyone else has run into and resolved the same issue. I am running ArcCatalog 10.6.1 and looking to manage fields for some feature classes and shapefiles. Going into the feature class properties and Fields tab I am able to see all of the fields and add new fields, but cannot delete any. There is not even a grayed out option anywhere, it just appears as if there is no option to delete fields. This occurs in feature classes in versioned databases, local file geodatabases, and shapefiles in a folder.
In the Data Management geoprocessing toolbox, there is a Fields/delete field tool, which is much easier to use and which will give you errors in the Results tab if it cannot run for some reason.