Not only will they find out everything, but they expect you to tell them everything too. Don't rely on the fact that an arrest or conviction has been "expunged." Tell them about it anyway, because they can get that information in most cases. It might seem like splitting hairs, but when the questionaire asks "have you been arrested or convicted....?" it is NOT asking "are there any arrests or convictions on your record?" The difference might seem like semantics to you, but it can bite you. You are expected to tell the truth regardless of whether it will show up on your record. Failure to disclose a conviction or arrest is more serious than whatever offense you got arrested for in the first place. So, you're better off listing everything.
Depending on the type of clearance, they may investigate you by going out and interviewing people who knew you years ago. An arrest can get expunged from your police record, but not from the memory of the guy whose nose you broke.