AC Single Phase Motor

One of the most common reasons for motor failure is broken windings. In this failure mode, the wire in one of the two windings has severed, and the magnetic field is not being generated. The obvious symptom of a broken winding is that the motor does not turn when powered, but it may spin freely when turned by hand.
Often, the failure is at one of the connection points, but the windings can fail anywhere. Unfortunately, if the winding is broken in the middle, it can be hard to find and is rarely worth repairing, as the motor will need to be rewound. If the winding is broken at one of the terminals, this can typically be repaired.
Testing for a broken winding is notoriously difficult on single-phase motors. Since the start winding contains a capacitor and has a different impedance, it is impractical to measure each winding to locate the fault.
All single-phase motors have two wires entering the casing, L and N. Often, there’s a third ground wire as well. Place a multimeter across the L and N leads and set the meter to resistance. The value should start at a lower resistance, but not zero, and should quickly increase a bit and level off at a slightly higher value (this is intentionally vague, as the exact resistance values depend on the motor size).
This action indicates that the capacitive start winding was indeed being charged properly and is functional, and since the resistance reached a steady level, this also indicates that the run winding is functional. If the start winding is open, the resistance will not change at all. If the run winding is open, the capacitor will keep charging and the resistance will slowly keep increasing.

The opposite of a broken winding is a short circuit where the insulation has failed. This might be found through visual inspection where worn, melted, or otherwise damaged insulation in the coil windings. However, sometimes the damage is hidden inside the casing, or invisible to the human eye, as certain chemical exposure or thermal degradation of insulation can lead to the change in dielectric properties of the wire insulation, which must be determined using other means.
The first test for determining insulation failure is to use a multimeter to see if there is continuity or low resistance between the S terminal and the housing or chassis. This test is then repeated with the R terminal. Finally, the C terminal and the housing should be tested. In spite of common language (common, grounded, etc), there should be high resistance between the C terminal and the motor chassis. If any of these tests indicate a low resistance, the insulation may be damaged.
Another test is to use a megohmmeter to test the insulation, in case the resistance value is lower than it should be, but not completely zero like a true short-circuit.
A megohmmeter places high voltage across its terminals so that very high resistances can be measured. Even if a motor has passed the multimeter test, it is still a good idea to test with a megohmmeter. Readings below a megaohm indicate either a direct short or an insulation or dielectric breakdown. Extra caution must be used when testing with a megohmmeter, as they can be used to apply 5000 V or more.

I (the author) first experienced single-phase AC motor overload when he was a kid. From the top bunk of his bunk bed, he kept piling clothes on the ceiling fan and turning it on. It seemed like a fun game, for a while. Eventually, a pair of pajama pants got wrapped around the fan, and combined with the load of the other clothes and the imbalanced nature of the stacking, the fan stopped spinning, humming lightly instead. Suddenly the lights went out in the room. The motor kept drawing more current until the breaker tripped.
Lots of conditions can lead to motor overload, such as physically overloading the motor, as described above, not performing maintenance, so that lubricated surfaces become unlubricated, collecting dust and debris that can stall the motor, and other such situations. Furthermore, many of these motors also have a thermal switch that can open if the motor is running too hot. This is sometimes a problem with home appliances when their vents are blocked on crowded countertops or shoved into a corner for aesthetic rather than functional purposes.
Overloading can be fixed by examining the motor and its operating conditions, and then changing them. Replacing a set of worn bearings, removing some of the load, applying grease, or moving the motor to a location with better ventilation can fix this problem. However, it might be worth checking the windings if the motor has been overloaded multiple times to ensure that the insulation has not been thermally damaged.

Some motors have a large capacitor for boosting and shifting the current to start the motor. This startup capacitor is only engaged, charging and discharging during startup. If a motor is equipped with a startup capacitor and intermittently or permanently fails to start, yet the motor turns freely, and no breakers are being flipped, it is possible the startup capacitor is bad.
Capacitors that have failed may be physically damaged. Any capacitor that is bulged at the sides or has leaked electrolyte should be replaced. However, some simply dry out, and there are no visible clues to their failure. A good multimeter can be used to check the startup capacitor. Take caution, however, when handling capacitors. If a capacitor is working, it may retain a charge for years after being disconnected from the power.