Mark Steen has a great post at the new OrangePhilosophy on how to catch plagiarists. I've used most of these methods (and found cases like most of those he gives), and they usually are good signs worthy at least of a Google search. Some of the cases I know about would be really funny if I didn't know that someone really thought they could pass off what they handed in as their own work.
The most obvious one was when someone handed in a paper that was pretty mediocre throughout but then had a paragraph from Kant's Critique of Pure Reason at the very end (and Kant wasn't course material). There's no style more distinctive than Kant (except perhaps Aquinas or Spinoza), and you have to be pretty stupid to think whoever is grading your paper could think today's undergrads could write something as obtuse and technical as Kant (well, the obtuse happens all the time, but not in the way Kant is obtuse).
I had someone personally give me a normal-seeming paper most of the way through, but there was one paragraph right in the middle that sounded like it was written by a graduate student in neuroscience. Those are probably the worst cases I've seen, but there are many others that were fairly obvious once you understand the kinds of things Mark discusses.