It may help to back up a step and analyze the meanings of the words they are derived from to understand when to use each of these adverbs.
1) Allegedly means 'according to what is or has been alleged' (i.e., declared without proof). In the USA, it's most frequently used by reporters when they want to cast aspersions on someone without being sued. E.g., The suspect allegedly stole a painting valued at $10,000,000. (Perhaps the suspect did it, but we don't have enough proof to accuse him directly.) Related words: alleged, to allege, allegation
2) Supposedly means 'according to what is generally assumed or believed.' It's often used to show that the speaker doubts the truth of the supposition. E.g., The suspect supposedly stole the painting without anyone on the inside helping him. (But, I believe someone working at the museum helped him.) Related words: supposed, to suppose, supposition
3) Reportedly means 'according to report or rumor.' It is used if there was an oral or written report made that is now being cited. E.g., Reportedly, the subject has fled across the border. (We are aware that immigration has a record of someone similar to the subject crossing the border an hour ago.) Related words: reported, to report, a report
4) Reputedly means 'according to reputation or general belief.' It is often used when someone or something's reputation is involved. E.g., The painting that was stolen was reputedly the finest work of artist X. (The reputation of the painting is involved.) Related words: reputed, of repute, reputation