Should you check YES or NO if an application askes if they may contact your current employer?

Answer

You can give permission to contact your present employer only if an offer is made. Advise employer you will not permit prior to an offer being made, but certainly afterwards. This protects you either way.

I'd say it depends on your situation. If you're working part-time while hunting for a full time "real" job, your current employer should be (obviously) aware that you NEED a full time job and should be nice enough to give you the reference. And always tell your current employer (if you're part time) that you intned to keep the PT job whether you intend to or not. But be courteous, always give two week's notice lest you be damned to hell for all eternity and lose that potentially good reference.

All depends on your relation with the current employer , if you go out to get pissed couple times a week with him, then let your potential employer contact him.