Unless you have a special application that requires a very specific kind of aggregate, what type of aggregate you get most often depends on what is economically available in a given region. Specifications normally specify aggregate properties that can be met by wide variety of aggregates. Normally, it's best to start with ASTM C-33 or your local State DOT specs (in the USA, equivalent agencies elsewhere), evaluate the spec, and then just specify exceptions or additions to the standard requirements as may be needed for your specific application.
If ASTM C-33 is used, sometimes local aggregates may not qualify under the grading requirements. However, within C-33 there are exceptions to the grading requirements such that an aggregate may be considered C-33 compliant even if it does not strictly comply with the C-33 requirements, but does have a satisfactory service record and is in compliance with the grading specifications of local public works specifying agencies.
Mix designs may not (usually don't) indicate the type of stone unless the type is specified, but do indicate test values that demonstrate compliance with the performance criteria specified.