I have just bought a second hand car, and I have the feeling that the clutch is worn out.
How can I do a quick clutch sanity check?
With the vehicle stopped, and in a safe place, start out in 4th gear, and slowly release the clutch while giving the engine gas.
If the clutch is in good condition, the car should stall - if there's any slipping, you'll probably feel it, and should disengage the clutch right away to prevent further damage.
If the car happens to start going without stalling, then chances are that the clutch is good as well.
There are two already great answers to this, but for future readers:
4th Gear Test
Put the vehicle in to forth gear, as you release the clutch give it gas as if it was a normal take off. If the vehicle stalls, chances are the clutch is still good. If the vehicles moves the chances are you have some level of slippage.
As Parker pointed out, you can perform the overtake test; when on an open highway (please ensure there are no other vehicles around, and that you're not in an area that is heavily populated), put the vehicle into its highest gear and press the gas pedal towards the floor as if you were going to pass a slow moving vehicle in front of you. If the RPMs go above what you expect them to be at (for example, in 5th gear at 80MPH, my RPMs usually sit around 3500 to 4000) then you have slippage.
This can be done from a stop, or while moving.
While Moving: While driving your vehicle, find a road with a decent hill and attempt to maintain a constant speed throughout the span of the hill. If you come from a flat surface, onto the hill, and your RPMs begin to climb in order to maintain speed (with bad slippage you'll begin to lose speed either way depending on the angle of the hill), then your clutch is slipping.
From a Stop: You can test your clutch this way, the same way you would the 4th Gear Test above, with two exceptions. You should be in 1st gear and, your car should not stall (unless you're just naturally bad with hills), if your clutch is slipping, the vehicle will roll backwards instead of moving forwards. Pressing harder on the gas can eventually move your car forwards as you climb higher in the RPM band.
There is another method called the tree test, but I wouldn't recommend this as it can damage your vehicle.
From the technical side, the motor has its maximum torque at a given RPM (read your cars documents). If the clutch slips only a little, it would slip at that RPM and max throttle.
The 4th-gear test has the drawback that it is difficult to check for slip at that RPM and full throttle, plus this puts enormous wear on the clutch, plus you'll only find out if there is much slip, but not little or none.
Finally, the key is to have a condition where flooring the gas doesn't change speed that fast, otherwise you won't notice that RPM and speed don't match. This means a high gear, maybe uphill in 3rd. The car must be at a speed where the RPM is at the max torque point, or just below. If you now floor the gas, and RPM stays stable, i.e. increases at the same rate as speed, then the clutch is OK.
As question from the comments about determining slip from OBD or similar: I'm not aware of such a feature directly (but aren't a pro of this), but you can read out RPM, speed and throttle. Apply a filter for data with high throttle, and draw RPM vs. speed. You should get as many straight lines as your car has gears, each with a different slope (speed RPM ratio). If there is slip, you should notice some datapoints above that straight lines.
(I think, I'll try that out)
Park on a steep hill with engine off. Put vehicle in 1st gear with hand brake off so that the engine is holding the vehicle in position. Depress the clutch to see how much travel there is before the vehicle moves.
If hardly any then your clutch will not last much longer. The further you have to depress the clutch to disengage the better.
This does not cause any further stress or damage as the running tests do and is safer.