Canonical Supernatural Abilities of Tolkien's Elves
Note: This reference is probably incomplete. Tolkien’s concept of his world evolved over time, so references which are bolded(from works written before 1937, when the Hobbit was published) shouldn’t be taken as the author’s final intent.
Elven musicians “can make the things of which they sing appear before the eyes of those that listen.” (Appendix A, Return of the King).
At least one Elf learned to speak the tongues of ‘birds and beasts’ (Celegorm, The Silmarillion, Of Eldamar and the Princes of the Eldalie).
Elves can count the number of people riding in a close formation, and notice their hair color and height, from a distance of at least seventeen miles(28 km). (Legolas in The Two Towers, The Riders of Rohan).
Elves who have been to Valinor glow (Glorfindel, Fellowship of the Ring “Flight to the Ford”, Galadriel, Return of the King “The Gray Havens”, Gildor, Fellowship of the Ring “Three is Company”).
Elves can walk atop the snow (Legolas, Fellowship of the Ring) and have remarkable cold tolerance (Legolas, Fellowship of the Ring, and the host of Fingolfin, The Silmarillion).
At least one Elf could influence by magic the direction paths take within his kingdom, so as to trap someone there. (Eöl, The Silmarillion, Of Maeglin).
Elves can read minds and communicate with each other telepathically (briefly referenced in Return of the King, “Many Partings”, expanded on in ’Osanwe-kenta’ in Vinyar Tengwar)
Elves can duel with song and knock down buildings with song (Finrod in The Silmarillion, Of Beren and Lúthien, Galadriel in Of The Rings of Power and the Third Age.)
Elves can heal with song (Lays of Beleriand, which credits Lúthien’s healing song as one “that Elvish wives long years had sung in those sad lives of war and weapons”)
Elves can conceal/temporarily cause to disappear possessions of theirs which they want to disguise (Lays of Beleriand, where Finrod uses magic to conceal the Ring of Barahir and other items which might identify his party, and to conceal them as Orcs).
Elves can create magical items: rope that unties itself when no longer needed, cloaks that make the wearer nearly invisible, rings that make mortal bearers immortal (though not spiritually and at a spiritual cost), the palantiri, gems that emit light, swords that glow in the presence of enemies and swords which are sentient and can talk. It’s unclear if Galadriel’s mirror is a magic item or if she has a magic ability to see distant and future things in it; she herself cautions the Hobbits that thinking about Elven abilities as magic is a mistake (The Fellowship of the Ring, “The Mirror of Galadriel”).