It appears that the active stoplist in use has been removed from the GUI in later versions of SQL - so the correct answer from Sem is now outdated. Nowhere in the SQL Server Management Studio can I find which stoplist is in use for a particular Full Text Catalog.
After quite a bit of digging, the following query will easily provide which stoplist is used for each fulltext catalog:
select so.name as tableName, sfc.name as fullTextCatalogName, sfi.is_enabled, sfi.stoplist_id, sfs.name as stoplistName
from sys.fulltext_indexes as sfi
left join sys.objects as so on so.object_id = sfi.object_id
left join sys.fulltext_catalogs as sfc on sfc.fulltext_catalog_id = sfi.fulltext_catalog_id
left join sys.fulltext_stoplists as sfs on sfi.stoplist_id = sfs.stoplist_id
So if stoplist_id is 0 - this indicates that this catalog is using the "default" system stoplist. If stoplist_id is NULL, this indicates no stoplist is in use (i.e. ALTER FULLTEXT INDEX ON {{TABLENAME}} SET STOPLIST = OFF).
And as indicated in another answer - if you want to additionally list WHAT stopwords are in the default system stoplist for a given language (assuming English here), you can:
SELECT * FROM sys.fulltext_system_stopwords WHERE language_id=1033
... see list with ID's of user defined stoplists:
SELECT * from sys.fulltext_stoplists
... and if you want to see what stopwords are in a user defined stoplist:
SELECT * from sys.fulltext_stopwords where language_id = 1033 and stoplist_id = {{a_valid_stoplist_id}}
I hope this helps, as I had to fix some of this on my application - and started really scratching my head trying to find where the active stoplist for an index was located - as I was used to just right click -> properties as worked in older versions of MSSQL...