Is it advisable to email director of graduate studies inquiring about the application status?
From my experience, you may if you think it is long overdue, as in if the expected deadline for your school "xxx" to send out all offers of admissions has elapsed. If that is the case, then it is okay to send in an inquiry email regarding the status of your application. 

Most schools usually send out the first round of admissions to very outstanding applicants first, then in that order. Also, domestic students often receive their admission offers first before international students do. These are the most obvious graduate school admission process but other applicant-unique factors may account for the delay in hearing back from your school. So you may have to factor all these in to see if you should have heard from your school or not regarding your application status. Contacting them before the given deadline for admission offers to end may paint a rather not so good image of you to them.

In my case, apparently the school had received more applications that year for that program than anticipated and so the selection process was rather complicated. I waited for the given time (month) all admissions (both international and domestic) to elapse before I contacted the graduate secretary. They immediately replied explaining the difficulties they faced that given year and the status of my application. Apparently I was placed on their waiting list. I hope this helps :)