Adafruit motorshield broken?
Without more details about your circuit it's difficult to troubleshoot your setup. But I'll give it a try. Bear in mind that I'm a newbie, so I cannot garantee that my advice won't burn your motors, your board or yourself. Follow my suggestions at your own risk. Also, always follow safety instructions and use safety equipment. Avoid dangerous voltages and avoid connecting your circuits directly to the mains. Instead, use good quality wall warts or low voltage batteries. That being said, here are my suggestions:


Try to power the motors without the Arduino and the motor shield, in a breadboard first. That will reduce the number of variables you will have to control. I assume your motors have 2 wires instead of 3 or 4. In that case, connect the wires to a battery rated for your motors (i.e., same voltage as your motors). This will asure you that you motors are working.
While your motor is hooked up to a breadboard, measure the current it is taking from the supply with the digital multimeter you are going to get. Then, hold the motor (prevent it from spinning) and take note of the current reading in the ampmeter. That will be your motor stall current.
Then hook up the motors to your motor shield. Make sure you are using a power supply that is capable of delivering the stall current you measured in step 2. Otherwise, your motors won't start spinning. If you can't get a more powerful supply, try to spin the motor yourself, with your hands. That may get them started with a lower current peak.
Make sure you power your motor and the Arduino using separate power supplies, with both grounds connected. That will help prevent power shortages from booting your MCU. Here is a link that tells you how to wire two power supplies to a set of motors (skip the first half of the page which is in Portuguese, the bottom half is in English). 
Your setup may be triggering protection circuitry of your power supply. Check out this answer for more details.