With the engine off, inspect the fan blades carefully. Make sure the blades are all in good shape and secured to the fan hub. Replace the fan if necessary.
Look carefully around the fan clutch for signs of oil leakage. Fluid-coupling and thermostatic fan clutches are filled with silicone-based oil; if you find a leak, replace the fan clutch.
Grab the fan assembly with your hands and move it back and forth. If the clutch or fan have excessive play, remove the fan and clutch and check if the water pump shaft -- where the fan-clutch assembly mounts -- displays excessive play. If the water pump shaft checks OK, replace the fan clutch.
Try to spin the fan with your hands. A healthy fluid-coupling clutch should offer some resistance to fan rotation, because the silicone oil is thick and viscous. If the fan spins without resistance or is hard to rotate, replace the clutch.
Turn on the engine and let it idle. If you have a thermostatic clutch type -- look for the thermostatic spring at the front of the clutch -- the fan should not turn until the engine reaches operating temperature. If the fan rotates when you turn on the engine, or the fan does not work after four or five minutes when the engine reaches operating temperature, replace the clutch. Keep an eye on the temperature gauge indicator on the instrument panel.
Test the fluid-coupling fan clutch visually with the engine running. This type of fan is designed to slip at high engine speeds. Have an assistant rev up the engine. At high speeds, you should see the fan slowing down or stopping.
The first sign of a fan clutch stuck in the permanently engaged position is a fan that never shuts off. Wait for the engine to cool down to dead cold and start it up; most fans will turn some whenever the engine's running, but they shouldn't be running hard enough to hear from inside the car with the hood shut. If you hear the rush of air from your fan while the engine is cold, there's a good chance the clutch is shot. If this is the case, you may also notice a drop in acceleration and fuel economy as the engine expends its energy unnecessarily blowing hot air.
A clutch that is worn out and won't engage will cause the engine to run hot, particularly at idle.