Should seemingly arbitrary things like "play piano for Church" or "intramural badminton" go on M.S. or PhD applications in sciences/engineering? And if they should, I assume that these things should go on a resume under something like "Activities and Community Service". I mean I consider being able to sight read music to play at my Church as a significant feat (how many people can do that?) and might sway one person on an admissions committee if he or she enjoys music and understands how hard one must have worked to accomplish this. 

Badminton?.... perhaps not so much
No.

These things are by and large irrelevant. Everyone has hobbies, and many of these may be challenging, but the purpose of a graduate school application is to understand your background as a potential scientist.

At best, this will make someone go "Huh, neat, the oboe..."

At worse, it will make someone assume that you're either attempting to pad out a relative lack of accomplishments, or don't really know what you're doing.