I was a TA for a graduate level class at one of the top universities in the US and I've had some interesting encounters with plagiarism.

I. The time I got caught for "plagiarizing".
In an intro systems class, me, a CS major, and my roommate, who wanted to minor in CS, were working together and I was "showing him the ropes". He was an intelligent student and we never worked together on the homeworks aside from general verbal discussions on what the solution could be. He used a Windows laptop and for one of the assignments, his C code wasn't compiling because he was missing some libraries and he told me he couldn't figure it out and we were approaching a deadline and asked me to compile it for him and send him back the binary. I did so, but when sending back the binary, in a rush, I accidentally mistook my HW folder for his (we'd downloaded this as a part of the assignment, and the folder structure was identical) and sent him my binary by mistake. Both of our solutions worked.
Obviously, we got "caught" in the most naive way. Our binaries had the same MD5 hash and the CMS flagged us. We were both confused at first, and then we realized what happened and explained it to the professor.
The proof was simple - just compile my roommate's binary and run it. However, he annulled our assignment to 0. We still both got As (because you could drop one homework) and while some may claim this was a gentle slap on the wrist, it felt unjust. We clearly made a dumb mistake and we shouldn't be punished at all, especially when we knew how rampant actual plagiarism was.

II. The time I caught students for "plagiarizing".
As Kevin points out in his post, there aren't really any incentives to catch students for cheating. As a TA, I get no benefit, and moreover, there's a cost. No one wants to be known as THAT TA who busts kids for using "a little help". Keeping that in mind, I was usually very lenient when it comes to cheating. I've noticed signs, but there was never enough proof to warrant the effort of calling someone out.
However, at one level it went too far. Two students who were partners for the "projects" had submitted nearly identical solutions for a complex Graphics homework assignment. They got the answer right, but I looked into their working and they both said "(9/5) / (4/3) == (4/7) / (5*9) = 1/3". I don't remember the exact values, but it was two steps of non-sense numbers and then a correct answer. I ended up reporting the case, mostly because I felt like my intelligence as a TA had been insulted. Are you seriously going to submit random numbers with a correct solution hoping I won't see? In any case, it didn't go anywhere.

III. Discovering a cheating ring.
At our university, one of my good friends and project partners told me there was an "enormous Asian cheating racket" - not to call out any specific race, I'm Asian too. I wasn't surprised - to be blatant, it made sense. We're very grade oriented with tiger parents. Then I learnt the extent of it. There were apparently Chinese forums and "outsourcers" you could send your homework problems to and they would solve it and give it back. In addition, there were special shared systems like DC++ where you could discover answers to homeworks for different classes at my university as well as Prelims, Midterms and Finals contributed by students of previous years. I was in shock. Students would leave exam halls to go to the bathroom just to look at these answers mid-exam. But was I gonna tattle? No.

IV. The reality at universities.
Not just in CS, but in every other subject, almost everybody cheats. Excuses that go around are:
"I've worked on it with someone else"
"Oh the TA in office hours told everybody the exact same solution"
"What? Cheating? me?"
"Maybe he/she took it from me, I didn't do it"

And look, people aren't stupid. We all know how cheating works. You get a homework assignment, and you re-write the sentences in your own language. You get some code from someone else and you define some useless functions with 1-2 lines of code. Or you arbitrarily re-organize lines of code. You rename all the variables. You re-organize your functions. You create some unnecessary classes.

There were students who distribute 10 homework assignments between 10 people (in groups of 2), and have one do the assignment (use office hours, friends, google, whatever) and the other literally re-write the assignment in LaTeX 9 different ways for the others to use. No one would ever really have to do the work.

The well known key to cheating is plausible deniability - if there's enough evidence you didn't do it, you didn't do it.