When do _WA_Sys_ statistics Get Updated? I have some of the auto generated _WA_Sys_ statistics in my database which have not been updated in a while (by comparison to other statistics in the same table)

The rule of thumb appears to be in tables >500 rows that statistics are updated at a 20% + 500 row change of data.

However, I can see using the following query 

SELECT  t.name,
        i.name,
        i.rowcnt,
        i.rowmodctr,
        p.last_updated
FROM    sys.sysindexes i
        JOIN sys.tables t
            ON i.id = t.object_id
        JOIN sys.stats s
            ON s.object_id = t.object_id AND i.name = s.name
        CROSS APPLY sys.dm_db_stats_properties(s.object_id,s.stats_id) p
WHERE   rowmodctr &gt; 0
ORDER BY i.rowmodctr DESC


That there is a table with a number of _WA_Sys_ statistics that are way out of date (and have a rowmodctr that is higher than 20% + 500)

If a run a query against the table and add one of the columns associated with the out of date _WA_Sys_ statistics in the WHERE clause and check the updated date of the statistic, I can see it has updated.

If i run the same query with the WHERE clause again, the statistic doesn't update

It seems like the _WA_Sys statistics update when a query is run that will use them and they are out of date?
SQL Server only automatically updates the statistics it uses. It may not be actively using the _WA_Sys statistics for cardinality estimation, though it may load them as part of the decision.

To see which statistics are being used for your query, add this to the end of your query, and look in the Messages tab:

OPTION(QUERYTRACEON 3604, QUERYTRACEON 2363);

If your database is in a compat level prior to 2014, you'll need these:

OPTION(QUERYTRACEON 3604, QUERYTRACEON 9292, QUERYTRACEON 9204);

These won't tell you why it's not using certain statistics, though generally if there are statistics created with a higher sampling percentage on the same column, statistics with lower sampling percentages will be disfavored.

Keep in mind that statistics updates don't occur on modification, only when queries run that use them