Aspartame metabolizes in the body as 2 neuro-toxins...formic acid and formaldehyde. Aspartame consumption has been linked to many diseases and disorders, including Gulf War Syndrome. When exposed to high temperatures, such as those that occur in shipping, aspartame further emits another neuro-toxin, methane gas.
Aspartame is a methyl ester of the product of two naturally occurring amino acids. What this means is that when some of aspartame is metabolized (some of it is, but no where near all of it), that it breaks down into methanol (wood alcohol) and these two natural amino acids. Concerns over toxicity arises from the production of methanol because metabolism by the same path as ethanol, the alcohol in drinks. This pass produces the formic acid and formaldehyde which are harmful; HOWEVER, a very small portion of aspartame is present in anything that uses it (it is very, very sweet so you don't need much of it). Because of its small presence, and lack of anywhere near full metabolization, the methanol produced is minuscule.
The body is also not "substantially" vulnerable to methanol because methyl esters are quite common in a lot of the flavenoids (the chemicals mainly responsible for flavor) in foods. Esters, in general, are responsible for taste/smell in many foods. A small portion of the foods you eat with methyl esters get metabolized into methanol + X, but your body has a tolerance. Something like cyanide has a very, very low tolerance (much lower than methanol), but I'm too lazy to look up the exact numbers.
As for the methane gas, it is not a very toxic chemical. It's toxicity is on par with table salt, unless that's all you're breathing, then you're going through asphyxiation, not poisoning. The formation of methane gas from aspartame by thermal degradation would require the absence of oxygen, otherwise it would likely combust into carbon dioxide and water.
The sweet taste stimulates the release of insulin.
That's definitely not how I learned how blood glucose regulation works. The sugar has to hit the blood to stimulate insulin.
A lot of opinions against aspartame are based on anecdotal evidence, but most consumers of aspartame have no ill effects. Of course some may be sensitive, but the level of hysteria that seems to be promoted is as if this effects everyone, but there has been no evidence for this yet. Most people would like further investigation into the toxicity of aspartame, which is a good route to pursuit, but thus far there has been no conclusive evidence for aspartame being a even a slight toxin, merely hopeful (if you could call it that) speculation.
A couple of funny things (one of which someone else pointed out), is that some studies employ extremely large doses to produce any ill effects (much much MUCH higher than anyone would EVER consume). I think it can be as high as the order of something like a couple thousand diet sodas within 24 hours; just absolutely ridiculous proportions. Many things in such huge doses can cause problems (i.e. salt). The second thing is that big list people like to through out with the symptoms. Those symptoms are all over the place. I have my suspicions that when they were compiling the list, they were merely finding people with these symptoms, which may or may not have already been explained by their current condition, and included these symptoms if they frequented diet drinks or other artificially sweetened foods. If any of these symptoms were even slightly common, you'd see many more f***ed up people.