I have two of these, one store bought and another made by a friend (guess which is my favorite). Mine also has velcro tabs so the bag comes off and you can take it to the trash. I never thought of using it with a glass in it so I have never used it for trash, I keep my small scissors and seam ripper in it so it is close by. Now I will put a small plastic glass in one of them and I can use them both.
Thanks for the idea.