I use an oscilloscope at work, but so far I've gotten-by without one at home. (I have brought one or two home-projects into work a couple of times to use the 'scope.)
If I were buying one, I'd buy a "real" benchtop oscilloscope capable of at least 20MHz (for the Arduino clock). That would cost a few hundred dollars. There are too many limitations to a little gadget that connects to the computer, and you've got to connect it to the computer, start & configure the software etc. It's "nice" if the computer can connect to the computer too, but I want it to work by itself. (Our Tektronix 'scopes at work don't have a computer interface, but some have a floppy drive, and the other a socket for a thumb drive so I can capture images.)
Typically, I'll use the 'scope at work to check for the presence of a clock (or sometimes to check the frequency) or to check data lines, address lines, chip enable lines, etc. You can't tell a lot from looking at address/data lines, but you can usually tell if you have an open or short or "no data", etc. Typically, I'm just looking for signal or no-signal, or high/low.