How likely is it that applying for jobs elsewhere could get back to my current employer?

[Blunt]

OK, long story short, I have a job that I'm happy at, but I think I'm long overdue for a promotion. Its gotten to the point where I think I need to start looking outside the company for opportunities, and I'm hoping to line up an interview or two to at least see what's out there. My game plan for now is to submit a resume or three to jobs posted online, but I'm definitely not making my resume public or anything like that. I don't want to jeopardize my standing with my current employer by having them find out I'm kind of looking for something else, because like I said, if they did resolve a couple issues with my current job description, I'd be totally content to stay with them indefinitely.

My question has to do with applying for other jobs, especially with staffing firms. If I pick and chose the spots I apply, to make sure I don't accidentally apply for something with my current employer or anything like that, am I relatively safe? Or, do these places share resumes at all? If for example, I applied for Job A with a staffing place, but they were also looking to fill Job B which happened to be with my current employer, would I run the risk of them finding out? What if I accidentally applied for a job with my current employer via a third party, would it be likely that they'd "tell" on me?

Any other advice for conducting a discreet job search? Anything I should do or avoid doing?