Mermaids are one of the most iconic creatures. But like many mythical creatures, mermaids are said to be biologically impossible (at least without genetic engineering). But just for the heck of it I'm going to present a scenario under which mermaids could have actually evolved
So What is a Mermaid?
Well, most often a mermaid is described as having the body of a woman and the tail of a fish, making them a chimera. Right off the bat this presents a problem for their scientific possibility. Women (being humans) are mammals and fish are an entirely different lineage of vertebrates and identical features in two lineages won't evolve twice any more than two languages that arose in different parts of the world will have the same vocabulary and grammatical structure. So are mermaids fish or mammals? Well, they clearly have mammary glands which would make them mammals. And furthermore they have opposable thumbs which would make them primates. so how could a fish tail evolve on a primate???
If mermaids are indeed mammals, then how did they get a fish tail? Well being mammals doesn't mean that one can't evolve fish like features. Take whales, they have very fish like shapes but they are still as much mammals as we are. Their tail simply has gotten much stronger and evolved a fluke at the end. But humans (and our closest kin) don't have tails, so where did the mermaids' tail come from? Well since we don't have tails, we instead use our legs to swim through the water, which may be the origin of a mermaids tail. Furthermore some babies are born with a birth defect where their legs are joined, resembling a mermaids tail.
And while we may call this a "birth defect" if a population of humans was very ocean, bound, it could help some of the people swim better and be an advantage. Furthermore, if you look at many depictions of mermaids, they often have knees.
Meaning what was once their legs is now their tail. But what about the fins at the end? Well that would have been their feet with each of the rays being toes. Now the tail is much more flexible than feet are so how could that be? Well it may be that the bones have turned into cartilage making it more flexible. The Sturgeon fished have cartilaginous elements of their skeleton that was once bone.
But What About the Scales (and Hair)?
Mermaids are often depicted with scales which seemingly presents another problem for their scientific probability because we don't think of mammals having scales. However, this simply isn't true. Rats have scale like structures on their tails and pangolins are covered in them.
The scales aren't like those of reptiles but are instead highly condensed fur. So it is possible that an aquatic primate could evolve scale like structures on its tail/legs. But that raises another problem: the scales on mammals (and their fur) are usually a shade of brown, red or blonde while mermaid tails come in many exotic colors. Most often blue but sometimes green and a very scarlet red, so how can this be if mammals don't have these colors? Well to answer that we can look to the sloth.
Some species have a symbiotic relationship with algae that live in their fur. The algae give them energy from sugars produced via photosynthesis while the sloth provides them with nutrients needed to grow. This can give them a green color. Similarly mermaids could have evolved a similar relationship with algae that live in their scales. The algae would also give their scales vibrant colors.
For example, Irish mermaids known as merrows are said to have a blue tail and green hair. So it may be argues that they have blue-green algae (cyannobacteria) living in their scale tales. And like the sloths, they may have green algae growing in their hair giving it its color. That would also explain why mermaids kept their hair. Most marine mammals loose it for streamlining but if it can be a source of energy from inhabiting algal symbionts then it would be advantageous for them to keep it. The algae may inhabit small pockets in the hair so a mermaids hair would appear white with no algae in it.
The reason people's hair turns white is because it has small air-holes in it. But when the algae inhabit their hair and scales, it would give it them vibrant colors. Their are four main color varieties of algae in the ocean: blue-green, green, brown (or golden) and red. So if a mermaid has blue hair, she has blue-green algae in it. If its green she has a green algae in it, if it's brown or blonde she has a brown algae in it and if it's red she has a red algae in it. It's the same with her tail.
But What About Mermen (And Reproduction)
A lingering question is how do mermaids reproduce and why are there so many more mermaids than mermen? Well there are several explanations for that. Now to get one thing out of the way: many people have said that mermaids can't reproduce with out visible genitals. But the problem with that argument is no visible genitals doesn't mean no genitals at all. For example whales have genitals like any other mammals but they keep them internally in a slit until they mate. So just because you can't see their genitals doesn't mean that they don't have them. But the question still remains why are there so many more mermaids than mermen? Well there are three explanations for that.
1) Mermen produce far more female sperm than males sperm so far more offspring will be female. This would explain why mermen are sometimes reported but far less often than mermen.
2) Mermaids are are sexually a-dimorphic; i.e. all individuals will look like females whether or not they are actually male or female. I.e. we've been seeing mermen all along and we just didn't recognize them as such. But just as some human females look like men, some mermaids would look like men.
3) Mermaids are hermaphroditic in that all possess both male and female parts. Hermaphrodity is rare among animals but occurs in very different groups of animals and there's no reason to think it can't occur in a mammal.
Well there's my take on scientifically plausible mermaids :P