Aspartame toxicity has been well studied by several groups.
Here you can find an interesting literature review, which covers also other sweeteners, such as saccharin and cyclamate.
Artificial sweeteners--do they bear a carcinogenic risk? - Weihrauch and Diehl, Ann Oncol. 2004
Owing to the existing studies, the following statements can be made about the carcinogenic potential of artificial sweeteners.
Saccharin induces bladder cancer in rats, when fed in high doses. However, rodents react to most sodium salts, such as sodium ascorbate, with urothel proliferation and neoplasia of the bladder.
Heavy artificial sweetener use (>1680 mg per day) leads to an increased relative risk of 1.3 for bladder cancer in humans. A more precise determination of the exact agents is not possible, because many artificial sweeteners are combined in current food products.
Despite unscientific articles in the mass media and scientific press, there is no evidence that the artificial sweetener aspartame bears a carcinogenic risk.
The approvals of new generation sweeteners (acesulfame-K, sucralose, alitame and neotame) are too recent to establish any epidemiological evidence about possible carcinogenic risks.
A more recent article indicates that DNA damage is increased by acesulfame-K and saccharin (more than aspartame), although none of those was mutagenic. Unfortunately I do not have access to the full-text, so I cannot really comment on the results.
Genotoxicity testing of low-calorie sweeteners: aspartame, acesulfame-K, and saccharin. - Bandyopadhyay et al., Drug Chem Toxicol. 2008
From the abstract:
The comet parameters of DNA were increased in the bone marrow cells due to the sweetener-induced DNA strand breaks, as revealed by increased comet-tail extent and percent DNA in the tail. ASK and saccharin were found to induce greater DNA damage than ASP. However, none could act as a potential mutagen in the Ames/Salmonella /microsome test. These findings are important, since they represent a potential health risk associated with the exposure to these agents.
There are several other studies and plenty of reviews and the general consensus is that there is no scientific proof that aspartame is carcinogenic.
As for multiple sclerosis I could not find one single paper linking MS to aspartame or sweeteners.
Aspartame has been said to cause pretty much any of the diseases à la mode of our time (cancer, MS, autism, depression, even death). This is mostly due to the preaching of some anti-aspartame groups that, alas, do not have any scientific proof to back up their beliefs. I bet the email you received did not have any references in it, did it?
This page on the Media Awareness Network is pretty enlightning on the matter.