This is NOT Purim Torah.
Dolfinim can give birth from humans. What are dolfinim? Says Rav Yehudah, mermaids.1
Bechoros 8a. Translation follows Rashi.
Whether there are actually mermaids is not for this discussion. Let's assume that there are.
Are mermaids considered animals that such a union would be a violation of bestiality, or would it be permissible?
Let's say that a human went ahead and did it anyway, and the mermaid gave birth to some weird mermaid-human crossbreed. Is the offspring considered human, mermaid, both, or neither?
In other words: is a mermaid human? Is the offspring of a human and a mermaid human?2
1Rashi describes a half-human half-fish creature called "sereine" in the French of his day. Sounds like a mermaid to me.
2Possibly relevant is the argument in Chullin 79a between R' Meir and the Chachamim regarding mating a mule with a mule. Granted that no viable offspring will be produced, will there be an issue of kelayim?
According to R' Meir, we go after the mothers: if both of them had a horse for a mother, for instance, it is permissible, while if one had a horse as a mother, and the other had a donkey as a mother, it is forbidden. According to the Chachamim, it is an independent species; we don't care about the mothers - it will always be permissible.