There are plenty of examples of elves (ex. Galadriel, Elrond, etc.) using magical abilities in LOTR. Even "lesser" elves than Elrond and Galadriel such as Arwen and Tauriel (in the movies at least) were shown using elf magic with athelas to heal others.
Were there any mention of Legolas using any elven magic?
Magic as it relates to the Elves is a somewhat cloudy subject. Galadriel comments to Sam,
This [the Mirror of Galadriel] is what your folk would call magic, I believe; though I do not understand clearly what they mean; and they seem also to use the same word of the deceits of the Enemy. But this, if you will, is the magic of Galadriel.
(Lord of the Rings, Book III, Chapter 7, "The Mirror of Galadriel")
Elves use a couple of different procedures that Tolkien has different names for:
Magic should be reserved for the operations of the Magician. Art is the human process that produces by the way (it is not its only or ultimate object) Secondary Belief. Art of the same sort, if more skilled and effortless, the elves can also use, or so the reports seem to show; but the more potent and specially elvish craft I will, for lack of a less debatable word, call Enchantment. Enchantment produces a Secondary World into which both designer and spectator can enter, to the satisfaction of their senses while they are inside; but in its purity it is artistic in desire and purpose. Magic produces, or pretends to produce, an alteration in the Primary World. It does not matter by whom it is said to be practised, fay or mortal, it remains distinct from the other two; it is not an art but a technique; its desire is power in this world, domination of things and wills.
("On Fairy-Stories")
It's not really clear from this description whether, for example, Galadriel's mirror falls under what Tolkien might call "Enchantment" or simply "Magic" proper. But neither, I think, describes anything that we see Legolas doing. He certainly has the ability to run quickly over snow without sinking or leaving much of a footprint; he is an extraordinarily good archer; he can walk a tightrope; he can see far distances with extraordinary clarity. These are all purely physical characteristics, though; there's no magic needed to produce those effects.
Other than that, we don't see him do much except follow and fight as he sees best.
Tauriel, of course, isn't present in any of the books, so there's no question of her doing anything; and Arwen has a much more passive role in the books (except, to an extent, in the "Tale of Aragorn and Arwen" in Appendix A)—I don't see any sense in which she uses magic either. We don't even know exactly the extent to which Elrond used magic in healing Frodo. We don't see exactly what goes on; though I suspect that the process was something like how Glorfindel began Frodo's healing:
He searched the wound on Frodo's shoulder with his fingers, and his face grew graver, as if what he learned disquieted him. But Frodo felt the chill lessen in his side and arm; a little warmth crept down from his shoulder to his hand, and the pain grew easier. The dusk of evening seemed to grow lighter about him, as if a cloud had been withdrawn. He saw his friends' faces more clearly again, and a measure of new hope and strength returned.
(Lord of the Rings, Book I, Chapter 12, "Flight to the Ford")
The Hobbit book shows potential enchantments used by Wood-Elves of Mirkwood (good power lingering in spots of their feasts that giant spiders did not like and it warded them off, putting sleep enchantents and maybe other things) of course the fact that Legolas didn't use those doesn't mean he can't, but it is not explicitly stated among his abilities. Throughought the curse of Lotr he only showed he enhanced elven perception of things:
"Much evil must befall a country before it wholly forgets the Elves, if once they dwelt there.'
That is true, said Legolas. 'But the Elves of this land were of a race strange to us of the silvan folk, and the trees and the grass do not now remember them. Only I hear the stones lament them: deep they delved us, fair they wrought us, high they builded us; but they are gone ..."
He also shows elven affinity with animals, communicating with them or easily mastering, that can be considered more like 'elven magic':
"A smaller and lighter horse, but restive and fiery, was brought to Legolas. Arod was his name. But Legolas asked them to take off saddle and rein. ‘I need them not,’ he said, and leaped lightly up, and to their wonder Arod was tame and willing beneath him, moving here and there with but a spoken word: such was the elvish way with all good beasts."
Also ability to communicate with trees and sense things outside of sensory perception:
"I do not think the wood feels evil, whatever tales may say,' said Legolas. '[...] I catch only the faintest echoes of dark places where the hearts of trees are black. There is no malice near us; but there is watchfulness and anger. [...] Do you not feel the tenseness? It takes my breath. [...] It is old, very old', said the Elf. 'So old that almost I feel young again, as I have not felt since I journeyed with you children. It is old and full of memory. I could have been happy here, if I had come in days of peace."
"Elves began it of course, waking trees up and teaching them to speak and learning their tree-talk. They always wished to talk to everything, the old Elves did."
Of course talking with animals and trees and other such things may not seem that impressive, but for Elves it is part of their innate powers, that humans regard as magic.
I'm not sure about what magical powers the other elves had, but Galadriel and Elrond had more magical abilities than the average elf. They both had elven rings of power. Galadriel had the white ring (Nenya) and Elrond had the blue ring (Vilya). These were rings of protection and had unique powers.
