I have heard that aspartame, the artifical sweetner used in some
sodas, has some negative health side effects.  What are the known
problems?  Please provide evidence/proof to support your answer. 
Thanks in advance.

There's lots of good info here:

http://www.answers.google.com/answers/threadview?id=322531

This claim has been investigated by TruthorFiction.com and their
report can be found there.

A summary of what I pointed out on the website which pinkfreud listed:

Aspartame is just two amino acids stuck together (which is
found in any meat, milk, or other protein products) with a methanol group.
As for the side effects, there are even more side effects reported
with placebo (inert) pills. There are tons of studies that gave huge
abuse doses of aspartame, and none found any unusual effects:

There was no change in urine metabolites from aspartame with the
equivalent of 10L of diet soda per day. There was no difference in
blood methanol concentrations between milk and aspartame. Aspartame
increases phenylalanine concentrations (thus the FDA warning only for
those who have phenylketonuria, a disease you're either born with or
not), but did not increase methanol or formaldehyde concentrations
beyond normal limits. A dose of aspartame in the 99th percentile
caused a slight increase in methanol concentrations, but that
concentration was within normal limits.

The documentation for this can be found on the PubMed research
database; the links are given in my original comment by going to the
aforementioned website above.

well its not to goo for you if its in a can of soda,left in the sun.
when the temp rises above 40c,it turns to ethanol,check out the troops
in quati,their diet cans were removed,as several soldiers fell ill
from ethanol poisoning

http://www.cancer.org/docroot/PED/content/PED_1_3X_Aspartame.asp?sitearea=PED
(very reliable site)

http://urbanlegends.about.com/library/blasp.htm
 for example see also: 
 http://www.mercola.com/2004/jan/7/aspartame_disease.htm
(questionable site)

http://www.holisticmed.com/aspartame/chapters.html
(questionable site)

http://www.ncbi.nlm.nih.gov/entrez/query.fcgi?cmd=Retrieve&db=PubMed&list_uids=3291200&dopt=Abstract
(formal scientific publication)

http://www.ncbi.nlm.nih.gov/entrez/query.fcgi?cmd=Retrieve&db=PubMed&list_uids=12180494&dopt=Abstract
(formal scientific publication)
CAVE: this one was written by NutraSweet (producer of aspartame)

Do not drink Diet Soda's.  I am 35 now, at 17, began drinking diet
coke (5-8 cans a day) My dizzy spells started about 3 months later.
They progresses to a quick loss of conciousness, and epilepsy. I had
mentioned 12 Years ago to my doctors that there may be a correlation,
but it was dismissed. Severe anxiety, depression, and manic depression
ensued.  I eventually had brain surgery for my seizures. (of which in
the hospital I was still drinking 4-5 diet cokes per day). I had what
I thought was a seizure while I slept (12 weeks ago) started to do
research on my suspicion of diet coke.  I stopped drinking diet soda,
my headaches stopped, I can think clear, etc.. I have classic
aspartame posioning. The same exact symptoms are found for aspartame
poisoning (exact- dizzy spells, seizures, aniety)  In the process of
suing Diet Coke. Don't drink it, it will be off the market within a
year

Pepsi is now working on a substitue for aspartame, and pulling it from
Diet Pepsi. They are doing this because they know Diet Coke is soon
facing so many law suites, that Nutra Sweet will soon be known as a
poison

in case you've never heard of it, there is a very good herbal
substitute for chemical sweeteners; it's called stevia, or sweetleaf.
Good extracts of stevia are now available in health food stores and
better supermarkets. It's supposedly more than 200 time sweeter than
sugar and (they say) actually good for you.  I grew a plot of it
myself very easily, but the homegrown variety has an odd aftertaste. 
The commercial extracts have managed to eliminate the aftertaste, and
they come in a clear liquid and a white powder form.  Nutrasweet and
other artificial sweetener companies tried to keep it off the market
for years.

I can tell you from personal experience that aspartame is dangerous. 
This is an "anecdotal" story, but true.  Beginning when aspartame
first came out, I quickly started consuming diet coke (around 82/83). 
My first "classic" migraine was shortly after that (blind spots,
flashing lights, followed by a severe headache).  I would have these
periodically, maybe once a month.  I did not associate these with
aspartame use, nor any of the other symptoms I began to experience
over the years.  I am now 51, and have not had aspartame since 96.  To
make a long story short, between 83 and 96, I became a heavy user of
diet coke with aspartame.  Over the years, I experienced the following
symptoms:
migraines, occurring more often with time
hip and joint pain
heart palpitations that were extreme to the point of being put on beta blockers
"restless leg syndrome" which made me want to jump out of my legs at night
panic attacks

None of the above symptoms did I associate with aspartame.  Then,
early 96, things started to go into what I call an "acute" phase for
me personally.  I began to have migraines two to three times a week. 
I had an "urgency to pee" that was really frustrating, infections were
ruled out by my doctor.  But the worst thing was, I began to
experience an extreme case of "brain fog" which has been well
documented as a symptom of aspartame poisoning (but i did not know
this at the time).  I felt exhausted 24/7, could barely get through
the day, and felt like I was drugged, groggy, not able to concentrate.
 I would of course drink more diet coke to try to "wake up."  I
believed I was dying, actually, but no one was able to find anything
wrong.  The doctors started to treat me like a hypochondriac, and I
was very worried.
At the time, I had a friend who had headaches from aspartame, and I
always poo poohed her claim.  I was so addicted to diet coke that I
was in total denial that it could be the cause of my problems.  She
suggested I get off of all aspartame, which I was reluctant to do,
because, after all, everyone said it was ok, and how could it possibly
cause ALL these problems.  No way!  But after so many doctor visits
with no help, I decided I had nothing to lose.  I spent a lot of time
wondering "what will I drink?" (a classic sign of addiction).  But I
did manage to quit, and amazingly enough, there was improvement. 
About 5 days after quitting, my family and I went on a camping trip
over the weekend, and I ran out of drinks.  Being not totally
convinced yet, I drank two diet cokes on Saturday night, and woke up
Sunday morning with the brain fog or grogginess.  I knew then, that
aspartame was responsible.

I have not had any aspartame in my system since 1996.  It is long
enough to know the long term results of avoidance of aspartame.  I do
not have migraines anymore.  Brain fog has NEVER come back.  No
urgency to pee, not heart palpitations (got off all meds), no panic
attacks, no restless leg syndrome, no joint or hip pain, etc.

I am a case study in aspartame poisoning and recovery.

I hope this helps people understand what they are dealing with. 
Aspartame is an incredibly insiduous, damaging CHEMICAL that has
multiple effects on one's body.  My symptoms were mine, other people
can have other manifestations, and they are well documented.  So, the
rule of thumb is, if you are a heavy user of aspartame, get it out of
your system for at least 60 days (watch your labels), and monitor your
progress.

I am a 23 year old diabetic type 1, being diabetic i obviosly cant eat
sugar so as most diabatics have surely done is turn to these sweetners
so ive been using diet soft drinks for like 5 years now and using all
the sugar free stuff, about 5 months ago i started to get really bad
migraines so bad that id have to lay down within half an hour or i
would throw up, then about 3 months ago i started to get this numbness
in my LEFT leg and my LEFT arm, now this scared the crap out of me
because whats one of the signs of a heart attack???? exactly! this was
going on every night in bed so i went to the doctor i had nerve tests
i had bone scans i had blood tests, nothing so i was referred to a
neurosurgeon, nope i refuse to pay that much money so i never went.
Now about 3 weeks ago i started a new diet of good foods absolutly no
'sugar free' products because my Blood sugar levels were absolutly
every where, since ive been on this diet ive had no migraines no
numbness in my leg or arm and i now no longer have breathing problems,
yesterday i received an email about aspartame poisoning and the side
effects and it makes sense that thats why i fell better for being of
it. SO you decide if your getting these problems try going 'cold
turkey' for a few weeks and you WILL notice the differance.