As a rule the employer is looking to see if your a felon, or just an all around screw up.
Some minor offenses can bar you from certain types of employment.

Like:
Domestic Violence / Care Giver
DUI / Truck Driver
Petty Theft / Cashier

Just depends what they are looking for. Also Public Records are just that, Public Records, anyone can look at them. For a small fee they will mail you a copy, if you know what it is your looking for.

The drug test is the easy one. Don't Smoke and it will never show up.