My Gaggia Classic packed up at the weekend so I decided to take it apart to diagnose the problem. I fixed it and I wish I'd taken some pictures now, because it was a simple debris blockage and was something that other people with Classics are likely to experience, simply because of the design. I'll describe it as best as I can for reference.
Symptoms:
Switch on the pump to pass some water through it. The pump makes all the normal noises but very little or no water comes out the portafilter. Take off the portafilter and there is still little to no water pumped through.
Diagnosis:
In the water container, there are two flexible pipes. As you face the machine, the one on the left draws water up and the one on the right is a pressure relief for the pump. Pull the right-hand tube out; if there's a blockage, the the pump will simply pump the water back down this tube.
Cure:
Unplug the machine. Take out the drip tray, the chrome overflow tube and the water container and lay the machine on its back. You'll see a Philips screw in the middle of where the hot water comes out into the portafilter. Undo it and pop the perforated disc off. Underneath is a round plate secured by two allen bolts. Take them out and wriggle the plate free. Both of these parts need cleaning thoroughly as they get clogged up with backed-up coffee residue and it's unlikely any cleaning or de-scaling products will clean them effectively.
Now look to the left of the two bolt holes in the machine. It won't be obvious if it's blocked, but there's a 2mm hole where the water is pumped through. You will need to clear this hole of scale to make the machine work again. I used a paper clip and loosened the scale, then stood the machine up and pumped some water through it. Quite a lot of crud pumped through into the drip tray.
Once you've done this, re-assemble the machine and it should work properly again.