I have some really stupid cheater stories, of course. Like the guy who handed in verbatim his buddy's assignment, including his buddy's name and student number at the top of the comments, and still kept denying it. Those are the easy cases to deal with and, while kind of funny, not very meaningful. If they fess up and apologize and co-operate you give them negative 100%; if they don't you give them an academic offence (which means automatically failing the course and automatically getting expelled from the university if they get a second offence).
There's definitely a lot more cheating going on than I can find evidence for, which I've come to accept is mostly out of my control. My only two real strategies for combatting it are to make the assignments as exciting/interesting as I can so that students don't mind doing them so much, and mandate that you need at least 50% on the exams to pass the course so that you can't get through just on the assignments.
The one that still affects me is a girl I had in a first-year course in my first year of TAing. She was one of the "cool kids" (a little slow transitioning out of her popular high school kid days, you know). Her friends sort of lived for partying and never took any of their courses seriously: most of them ended up failing the CS course. It was a lab scenario where it was scheduled for 2 hours, but if you finished your work in an hour (which many did), you left early. After all her friends had finished the bare minimum and had left, she would stick around, though, for an extra half hour or hour, asking questions and redoing the work she'd already done to make sure she understood it at all properly. She wasn't doing insanely well--I think she was carrying a mark in the course somewhere around the high 70s--but definitely better than her friends and she was putting a lot of work into what obviously didn't come naturally for her.
Then came her final assignment of the course. It was gorgeous! It was their only assignment where they had to incorporate object-oriented design (the course was taught in C++) and she did everything perfectly. It was definitely the best assignment out of the entire class, and we had a lot of really smart cookies in the class. I wrote glowing praises all over it. I ran into her at a bus stop a month or so later and gushed about how impressive her assignment was and how much work she must have put it into it and she just quietly smiled and didn't say much beyond "thank you".
Then about a year later I ran into one of her friends and the topic of her came up again. I said again how impressive it was and said "either she found someone else to write it or she put a lot of work into that". He got awkward and said "I don't want to get her in trouble, but let's just say she didn't put a lot of work into it".
It really did a number on me. I was at the end of my Master's degree at the time and it through me for a loop, wondering if I even wanted to come back for my Ph.D. Research is okay and all, but really the only reason I was in grad school was for the teaching. I eventually did go back and things have gone well, but it surprised me what effect one (previously) good student cheating can have on you :\