I think what they check varies. If they find something out of the ordinary they will do a more detailed check than if their initial search turns up clean. One sure thing is they will run a credit report, a school isn't going to want a student there that wont be able to pay for second year that would hurt their graduation % and also though its common these days you would think that it would not be a common problem for MBA students. They will run a check for a criminal record to see if you have anything there that concerns them. If you have 6 DUI convictions or an assualt and battery they may take issue with that unless you talked about it in your optional essay or addressed it at some point.
As for an employment check I think they will care whats your post college grad, they wont care if you worked at a grocery store in high school. So for not putting you worked as a waiter that shouldn't be an issue. As for putting an internship on your resume, schools usually state they only want to know your post graduation employment history...so unless it happened after college they probably wont even consider it. Internships typically aren't seen as valid unless its very unusual.
For references at most they will spot check them to see if they even exist. They may call one up and ask if they wrote a recommendation for you. I think with consultants and all the other knowledge people have these days on the app process, schools are trying to prevent people from scamming their way. Though it still does happen and you read stories about people doing stupid things trying to cover it up, like the guy who broke into his undergrad to change his transcript.