The Wikipedia entry on the aspartame controversy is a good place to start. It contains links back to the literature as well. Speaking as a biologist, I think that aspartame is safe for most people. If you suffer from phenylketonuria, you shouldn't eat it. If it triggers migraines for you, you shouldn't eat it. Other than that, go for it, and don't consume more than three or four grams a day. There is no solid evidence that it causes cancer, seizures, brain tumors, or any other disease. My personal opinion is that it's one of the safest artificial sweeteners on the market; there are strong a priori reasons to expect that it is safe, based on the molecular structure, and then there's the whole lack of evidence to the contrary.
Sucralose (Splenda) is one artificial sweetener that I'm wary of. Insufficient research has been done on the way and the extent to which chlorine atoms from the sweetener exchange into biomolecules in the body and modify them. It could be perfectly safe, or it could be a time bomb with a very long fuse. I'm not super-paranoid about it, but I generally choose products that don't contain Splenda.
I would prefer to drink soda sweetened with sodium cyclamate than aspartame, because it tastes better. Unfortunately, cyclamate sweeteners were banned in the U.S. over forty years ago, as the result of flawed science and bad politics. If you live in a country where cyclamate sweeteners are still legal, you might want to try them.