My favorite type is cartridges.
I like the ease of use, and the fact that you can save right on them (usually, i'm sure someone out there will find the proof to disprove my statement)
The down side is the fact that these are usually battery run, so when the battery runs out it's not good for much. The new ds carts fixed this problem, and i thank them very much for it. (maybe it's just with the flash devices.... again, i don't know)
DVDs are great because you can fit huge good looking games on them. I like this but I'm also really good at dropping and breaking DVDs.
UMDs are just like diskettes without the ability to have built in saves, blah. UMDs also only work in psp, and what good is that?
Whatever the other choices are are just crap.