For this scenario, you can try the following using core service as there is no built in methods for this:
UsingItemsFilterData filterCriteria = new UsingItemsFilterData();
filterCriteria.BaseColumns = ListBaseColumns.Id;
filterCriteria.IncludeLocalCopies = true;
var resultList =CoreServiceClient.GetList("SchemaURI", filterCriteria);
If you don't want to use or have core service the same can be achieved using TOM.Net API, there is a method on IdentifiableObject called HasUsingItems() which will give you the details also. Check this link once !!
You can have a look in this link which will give you ideas on how to use core service more effectively for similar kind of scenarios.
Let me know if this helps !!
