As with any equipment, do as much testing with the equipment disconnected first!
Switch your meter to Ohms, test across the primary windings, is there a reading? With the equipment switch in the on position, test across the mains plug pins, is there a reading? Test across the secondary of the transformer, is the reading low high or open? Swap your meter leads over.
Open = fault
near zero = fault
reading = may be O.K.
Yes, you can measure for the presence of 12V on the 200Vrange, but it may not be accurate.