No other consumable substance on the face of the earth has been studied more than aspartame. Over 30 years and over 200 scientific studies (the REAL kind, with words like "double-blind," "placebo-controlled," and "randomized" in their titles) have proven, re-proven, and reaffirmed, that aspartame is safe.
Researchers have spent a great deal of time reviewing every claim by people who believe that aspartame causes everything from MS (which, by the way, no one knows the cause, and has been documented for over 200 years, LONG before aspartame was even invented) to migraines and bulging eyes. Not a single claim of any cause has panned out under true scientific scrutiny. If you do have "proof" that aspartame causes any sort of disorder, look through your proof and see if there are any scientific studies that were done. Scientific studies use the big words above -- randomized, placebo-controlled, and double-blind. Those tenets of a true scientific study ensure that there is no bias in the group and that the substance is tested against a "placebo," or a control substance. In essence, if there is no appreciable difference between the group which took the tested substance and the group which took a placebo, you can say that the substance did not appreciably contribute to those people developing the symptom in question.
Now, to more of the scientific detail. Aspartame is a compound made of two ingredients: aspartic acid, and phenylalanine. When aspartame is consumed, it is broken down in the body into three components: aspartic acid, phenylalanine, and methanol. These three substances are then absorbed by the body and used in natural body processes, the same way your body uses those three substances when they are derived from food. None of the components of aspartame, nor aspartame itself, builds up in the body. The amounts of the three substances derived from aspartame are far smaller than you would get from food as well. For example, if you were to have a glass of tomato juice, you'd get about six times the methanol than you'd get from an aspartame-sweetened drink of roughly the same size. A glass of skim milk would give you six times the phenylalanine and thirteen times the aspartic acid you'd get from an equivalent aspartame-sweetened drink.
In short, hundreds of scientific studies over the past three decades have all resulted in the same outcome, and the same conclusion: aspartame is safe.