Aspartame, when broken down produces, supposedly, methanol (toxic substance). However, a Ph.D. chemist (who incidentally was my old organic chemistry prof) said that he thinks the whole thing is bullshit. He said it's ludicrous. I tend to believe him seeing as how people consume 10+ diet sodas a day and are still alive (and not blind from the methanol biproduct).
Aside from that, caffeine, usually present in modest quantites in diet sodas, is a diuretic, meaning it will dehydrate you. As a general rule I stick with diet sodas that are caffeine free (i.e.- diet caffeine coke, diet A&W root beer, diet rite, etc.)
Finally, there is discussion over the reflex caused in response to the taste of something sweet, known as "cephalic phase response". The bottom line is this: you taste something sweet, your body prepares for an influx of simple carbohydrates and ceases production of glucose from starches and proteins. When the body doesn't get the simple carbohydrates (diet sodas have 0 calories) you become hungry.
So... bottom line: I wouldn't drink more than one or two a day, caffeine free, and drink them during a meal if at all possible.
Hope this helps.