Use this:
SELECT  Closing_Date = DATEADD(MONTH, DATEDIFF(MONTH, 0, Closing_Date), 0), 
        Category,  
        COUNT(Status) TotalCount 
FROM    MyTable
WHERE   Closing_Date >= '2012-02-01' 
AND     Closing_Date <= '2012-12-31'
AND     Defect_Status1 IS NOT NULL
GROUP BY DATEADD(MONTH, DATEDIFF(MONTH, 0, Closing_Date), 0), Category;
Alternatively, you could use something like this:
SELECT  Closing_Year = DATEPART(YEAR, Closing_Date),
        Closing_Month = DATEPART(MONTH, Closing_Date),
        Category,  
        COUNT(Status) TotalCount 
FROM    MyTable
WHERE   Closing_Date >= '2012-02-01' 
AND     Closing_Date <= '2012-12-31'
AND     Defect_Status1 IS NOT NULL
GROUP BY DATEPART(YEAR, Closing_Date), DATEPART(MONTH, Closing_Date), Category;