When I was a respectable engineer, I used it for...well...everything. It's hard to even list it all because it's soooo damn much.
Now that I have my own business, and am doing analog audio stuff, I don't use it for much other than tracking down the occasional gremlin. For example, sometimes I can hear a little "hair" working it's way into the circuit. I want to be able to figure out where it's coming from. Stuff like this is generally simple enough that I can almost go from just thinking about a design, to producing a board, and 9/10 times the prototype board comes in and it does pretty close to what I need it to do the first time.
Even if you're a hobbyiest, though, anything that has to do with timing is prime stomping grounds for a scope. Communication, driving steppers, debouncing contacts, Class B or AB amps (crossover distortion, for example), tracking down inadequate decoupling (rails sagging or excess ripple), etc etc etc. Anything that happens faster than a couple of seconds and can't be viewed on a DMM, the scope comes out.
But if I just need to get something like a PIC going, and maybe it needs to take some serial input and drive something, I can generally do that with nothing but paying attention to the datasheet, and maybe a DMM to make sure things I want to switch are actually switching, and things I want to read for testing purposes are actually at the correct voltages. Once I know I can communicate with the outside world, things happen so slowly and have so much margin that a scope is not all that critical.