Is There A Way To Tell How Many Times A Query Has Executed? I am attempting to determine the number of times a particular query has executed in the course of a day. Is there a simple script that I can run to determine this?
You can use this query with few caution and need to understand thoroughly what the result is telling you.

sys.dm_exec_query_stats (Transact-SQL)


  Returns aggregate performance statistics for cached query plans in SQL
  Server. The view contains one row per query statement within the
  cached plan, and the lifetime of the rows are tied to the plan itself.
  When a plan is removed from the cache, the corresponding rows are
  eliminated from this view.



If a plan is not cached you will not see those query and does not
mean those did not run.
No way to aggregate accurately per day, you only get last execution time and number of times that the plan has been executed since it was last compiled.


Copied from above link and added 2 more columns.

SELECT qs.execution_count,
       creation_time, 
             last_execution_time,  
    SUBSTRING(qt.text,qs.statement_start_offset/2 +1,   
                 (CASE WHEN qs.statement_end_offset = -1   
                       THEN LEN(CONVERT(nvarchar(max), qt.text)) * 2   
                       ELSE qs.statement_end_offset end -  
                            qs.statement_start_offset  
                 )/2  
             ) AS query_text,   
     qt.dbid, dbname= DB_NAME (qt.dbid), qt.objectid,   
     qs.total_rows, qs.last_rows, qs.min_rows, qs.max_rows  
FROM sys.dm_exec_query_stats AS qs   
CROSS APPLY sys.dm_exec_sql_text(qs.sql_handle) AS qt   
--WHERE qt.text like '%SELECT%'   
ORDER BY qs.execution_count DESC; 


If you need an exact count for 24 hours for any query you will need to set up your own extended event collection.