On a job application, what if I didn't put down my social security number?

I was talking with a friend this evening and she mentioned that she doesn’t put down her social security number on job applications. There had been an incident where some employees of a linen store, in state, that had just thrown away old job applications. I didn’t really realize how easy it would be for someone to just get ahold of my social security number. I know it wouldn’t look well upon myself if I didn’t put anything down. But, I don’t want my information to end up in the wrong hands. Suggestions of what I could do?