Symbolism, Imagery, Allegory
Oskar first learns he can "singshatter" glass when he screams after his mother tries to take away his drum. His poor schoolteacher's eyeglasses explode when she tries to do the same. He often shatters glass just because he can—he destroys windows in buildings all over town. He uses this skill to tempt people into stealing from shop windows through small holes he "sings" into them. He puts it at the service of the Dusters and they go around terrorizing the city, breaking and entering wherever they can. So the glass-shattering is also a symbol of control, but it's more destructive control.
Oskar does see this ability as something of an art form. Bebra even hires him as a performer on the basis of his singshattering prowess. As Oskar gets older, he refines his ability. For example, he carves a delicate heart into a goblet for Roswitha. And he tends to avoid shattering beautiful stained glass windows in churches. He's an artist, after all.
Of course, all this glass shattering mirrors (no pun intended) a much more destructive event witnessed by Oskar: Kristallnacht, the "Night of Broken Glass" in November of 1938. Hitler Youth and SA squads roamed through towns in German and border areas, destroying 267 synagogues and about 7500 businesses. During the riots, about 30,000 Jewish men were rounded up and sent to prison camps—the first time that Jews were imprisoned just for being Jews. Soon after Kristallnacht, the Nazis made it illegal for Jews to attend German schools, practice their professions, enter German business establishments, or get driver's licenses. (Source).
Grass knew that Kristallnacht was the culmination of years of anti-Semitic propaganda that the German people were very aware of, and that it marked the beginning of the destruction of Jews in Europe. We think that's why he gave Oskar his glass-shattering ability—to remind his German readers about it.
But just as Grass minimized the symbolic aspect of the drums, he did the same for the glass-shattering. He described it as just something Oskar uses to protect his drum. He noted that it's very possible for singers to shatter glass given the right conditions, so no big deal, right? And just like his drum, he uses the skill first for protection, then as an art form, and finally, as a job with the circus.