There are questions as to whether the initial testing of aspartame was undertaken in the correct way. However, that was decades ago, and there is now huge amounts of evidence that apart from people who have specific allergies aspartame is safe when consumed in normal amounts.
Personally it took me some time to get used to drinking artificially sweetened drinks, when I was losing weight. And, I only became used to sucralose and acesulfame potassium. While I am confident that aspartame is safe, as I haven't become used to drinking it, I find drinks sweetened with it to not taste very nice, and hence avoid it for that reason.
From what I've read, aspartame has better proven safety than the artificial sweeteners I do consume.