Aspartame came out when my older DDs were small. I asked our pediatrician, who IMO is VERY conservative in his approach to medicine, if there were any dangers in using it. He told me that aspartame is a protien and that some people have allergies to some protiens (ie: peanuts) so that IF you have a sensitivity to this protien that you should not use it. Since we do not have that problem we have used it, and all of our 4 DDs have used it without problem.
Also, I have been diagnosed with Multiple Sclerosis - relatively benign thank God - and I asked my neurologist (who is a specialist in MS) about the aspartame claims. He said "That's horse (poo)! Drink your diet coke and don't worry about it."
I know that some people are very concerned about only putting things that are completely "natural" in their bodies. And that is fine. And if so, then you don't want to use aspartame. But for the rest of us, I think that we can pretty much rest assured that we are safe in using it.
(ALTHOUGH, I do believe that there may be something to the claims that using aspartame increases your hunger. So I am trying to use it very little these days. I use sweet n low and splenda.)