Formality of "悲哀" as opposed to "悲傷" and "哀傷"
“哀伤” is usually a noun, while “悲伤” and “悲哀” are usually adjective.
“哀伤”, “悲伤” both could mean severe sadness, and are often used in the context that one has lost someone he loved or something he valued.

任何语言都无法形容他失去亲人的哀伤。
No words could describe the sorrow of losing his family.
刚刚失去亲人的他看起来十分悲伤。
Having lost his family, he looks very sad.

Though “悲哀” could mean sadness in some context (like on a funeral), it mostly means “pitiful”, and is more often used to describe things rather than emotion.

人，失去了梦想，是多么悲哀的事情。
How pitiful it is, when one loses his dreams.
你为了金钱丧失了尊严，我真为你感到悲哀！
You lost your honor for money. I feel sorry for you.