Allegedly, supposedly, reportedly

These three adverbs are the closest equivalents to what is known, in French, as « le conditionnel journalistique » which is widely used both in the press and all over the modern media. This French verbal mode expresses a certain idea of doubt, ambiguity, or uncertainty.
Example: « Ma voisine, qui est un peu timbrée, aurait vu une soucoupe volante atterrir dans son jardin ». Here, the speaker doubts the veracity or truth of what s/he is saying. His/her doubts bear on the reliability of the source of information (here, a loony lady who probably hallucinated it all!).

This very particular form of conditional is never translated as would in English, so you must be very careful with the tense and mode that you use. The indicative mode is generally preferred. In most cases, to express this idea of doubtfulness, and in order to detach themselves from the information being given, English speakers tend to use one of the following adverbs, technically known as intra-sentential commenting devices – allegedly, supposedly/reputedly, and reportedly. Through them, the speaker indicates that s/he detaches himself/herself from the assumption, and warns that the source of information might not be quite reliable. 

Allegedly

According to the degree of unreliability, a different adverb will be used. For example, "allegedly" means that the source of information is unknown (i.e. you cannot trace it back to any known person, written report, etc.) and that, therefore, you consider the assumption to be highly doubtful. 
Example: "The Virgin Mary was allegedly transported in body and soul into Heaven" (the belief is not based on any clear source, and many people outside of the Catholic faith consider it as mere legend).

Supposedly

With "supposedly", however, the degree of uncertainty is lesser; you are dealing with general truths, things that are traditionally held as true, although they cannot be proven. There might be a source for the information, but that very source is viewed with suspicion. Example: "Christ supposedly died on the cross" (there is a written report, and most historians believe the story of the crucifixion is true, yet many people doubt the reliability of the story).

Reputedly

Another possible adverb here would be "reputedly" – meaning a belief that is widely shared and held as serious/true. Lastly, "reportedly" is the most neutral adverb of all four, and simply expresses the idea that "someone else said it". It is a way to neutrally detach yourself from the statement, without giving a hint of a personal opinion.
Example: "My neighbor is reportedly a ballet dancer" (i.e. I was told so, it might be true, but I’m not sure). So, now… how would you say « Ma voisine, qui est un peu timbrée, aurait vu une soucoupe volante atterrir dans son jardin » in English?