Is Aspartame Safe To Eat?

It's very controversial. And the short answer may be that although it's probably safe, you have to weigh up any benefits it may bring you (like helping in weight loss or preventing tooth decay) against the many allegations thrown against it (mostly hysterical in tone, and not very well researched, it has to be said).

Any government or reasonably objective website you consult will tell you that it's safe enough in reasonable doses.  Although there are plenty of scare stories about aspartame (marketed as NutraSweet), these are only stories; there isn't a big body of research that can consistently show any links between aspartame and cancer, multiple schlerosis, asthma, etc.

Here's a list of "aspartame is safe" advocates.

BBC
US Food and Drug Agency
Manufacturer's website
Nemour's Foundation
Group of Diet Food Manufacturers
A university student's essay that concisely sums up the validity of arguments for the alleged dangers of aspartame.
One British academic's assessment (more technical version of the last essay)