There are four main kinds of clutch problems: a slipping clutch, a dragging clutch, a juddering clutch and a squealing clutch. Let's look at each one in more detail.
A slipping clutch, an alarming and potentially destructive problem, occurs when the clutch plate doesn't experience enough friction to prevent it from slipping, even when the clutch is fully engaged. It will normally be noticed first when the car has to pull hard in a higher gear - the engine revs will rise while the car's speed doesn't change. The problem is caused by too little free play in the clutch actuating mechanism, a clutch plate worn beyond the limit of travel of the pressure plate, or by oil on the friction linings of the clutch plate.
A dragging clutch is the opposite of a slipping one. It occurs when the clutch plate, for some reason, is not fully released by the surfaces clutching it, when you depress the clutch pedal. Instead of coming to a stop while the flywheel continues to rotate, the clutch plate is then dragged along by the flywheel. This will make it difficult to shift from neutral into first gear when the car is at rest.
A dragging clutch can be caused by various things - a warped clutch disc, loose friction linings, a defective pressure plate not pressing squarely on the disc - but very often it's simply a matter of the full movement of the clutch pedal not being transferred to the release fork on the clutch. Perhaps the pedal has excessive free play, or, if it's a hydraulically actuated clutch, perhaps the rubber seals in the slave and master cylinders are leaking.
A juddering clutch will shudder, sometimes violently, on take-up. It has several possible causes: defective engine and gearbox mountings, allowing excessive rocking; worn rubber bushes on rear axle locating arms; oil on the friction linings; binding in the linkage between pedal and release fork; a clutch plate not sliding freely on its splines. The spectre of oil on the linings is the most worrying, but it's by no means the most common cause. Many older cars have a slight oil leak into the bellhousing past the rear main oil seal. As long as it stays slight, as it can do for years, it's unlikely to get on the clutch linings.
A squealing clutch is really a cry of distress from the release bearing. This robust thrust bearing should only be called upon to work during the fractions of a second when the clutch pedal is de- pressed-and-released, and then it will outlast the clutch. But some people are in the habit of keeping the clutch pedal depressed for minutes on end, while waiting at traffic lights for instance. During that time the release bearing is continuously working, treatment it was never designed for. Even worse is the habit of using the clutch pedal as a footrest in normal driving, thus keeping the release bearing working all the time. Maltreated like this, the release bearing will wear prematurely, and let you know about it.