Tonight at religious school we were discussing Kosher laws and how to build/keep a kosher kitchen. While we were discussing different meats, one of the jokers in school asked if you could eat a mermaid, since it would have scales and fins. Would it be possible to eat one?
The Torah says that we may not eat anything without fins and scales if it is among the "sea-life" or of the "beasts of the sea" (Vaykira 11:10) which the Ramban thinks refers specifically to sea creatures that have two legs like those of land-creatures. Assuming that he's talking about mermaids, he implies that they have two legs, instead of fins, and are therefore not kosher.
However, the mermaids that I've seen in Disney movies do not have legs, but do have scales and (I believe) fins. Raavad, though, in his commentary to Toras Kohanim (to the verse above) includes sirens (a.k.a. mermaids) in the list of non-kosher animals. As far as psak halacha, the Aruch Hashulchan (Y.D. 83:10) also states that mermaids are not kosher. His reasoning is that the Torah only permits fish that have fins and scales, and even if a mermaid were to have those features, she is not actually a fish.
As always, be sure to contact your local orthodox rabbi if this becomes a matter of practical concern.
The gemara in Bechoros 8a, which discusses mermaids, says they can be impregnated by people. As such, they would be non-kosher due to the qualification found there (Bech. 7b) a kosher fish lays eggs and a non-kosher fish births a live creature. This is the simple understanding found in the pirushim on the Tosefta (1:5) there.
Tosafos there complicates things for this discussion by bringing a gemara in Avoda Zara 40a where both kosher and non-kosher fish lay eggs, but at the very least it is the non-kosher fish there whose eggs are impregnated internally, so that would still work with the understanding here.
As per Matt's request to quote some poskim I'll quote the Machklokes brought in Darkei Tshuva 83:5. He quotes the Halachos Kitanos 2:5 who says someone brought him a humanoid fish with arms and legs and fins and scales, he ruled that it was not kosher as only creatures that look like fish with these signs are kosher. The D.T. goes on to quote a Shiurei Bracha who quotes this Halachos Kitanos and argues with him, setting out the position that any sea creature with fins and scales is kosher. The Darkei Tshuva does not side with either opinion, seemingly leaving the question of whether or not 'fins and scales' help to permit any and all sea creatures as a machlokes.