Inference (It Seems / Probably)

How sure you are: a bare possibility, or what you actually expect?

～だろう vs ～かもしれない: だろう is your expectation ('probably'), so it leans towards the thing being true. かもしれない only concedes that it might be, and works equally for a one-in-ten chance. Using かもしれない where you mean だろう makes you sound noncommittal.

You are basing the guess on something you can see or feel right now.

～ようだ vs ～だろう: Use ようだ. It reports an inference from evidence in front of you (雨が降ったようだ, looking at the wet road). だろう is a judgement from general knowledge and does not claim any particular evidence, so it sounds oddly detached when you are looking straight at it.

Passing on something you were told, in casual conversation.

～んだって vs ～ようだ: Use んだって. It is spoken hearsay and marks the information as second-hand and not yours to vouch for. ようだ claims your own inference from evidence, so using it for something you merely heard overstates what you know.

Relaying information without committing to how reliable it is.

～らしい vs ～ということだ: らしい is loose hearsay: you picked it up somewhere and are not standing behind it. ということだ frames the same content as a definite report from a source, which is why it suits news and official relay. らしい in a report sounds like gossip.

Raising a possibility you think the other person has not considered.

もしかすると〜かもしれない vs ～ということだ: Use もしかすると〜かもしれない. The もしかすると flags that what follows is a deliberate 'what if', which is what makes it land as a suggestion. ということだ relays a report you received and cannot introduce a hypothesis at all.

Formal writing: did you hear it, or are you inferring it from what you observe?

～とみえて vs ～とのことだ: とみえて draws a conclusion from evidence ('apparently, judging by...') and leads into the clause that follows. とのことだ relays what you were told and closes the sentence. The register is the same; what differs is whether you are citing a source or reading the signs.