The way Hermione described the ways to destroy Horcruxes implied that there were a few. Could you name the confirmed ones?
The books say that a Horcrux is destroyed if the container is destroyed beyond magical repair. (I don’t have a quote, but that gets the gist across.) There aren’t many bits of magic that can do this, but we do have a few.
Generally, each of these are as deadly and brutal as the sort of magic you’d need to create a horcrux. Just as creating a horcrux involves killing somebody else, so destroying one is killing part of somebody’s soul. There aren’t many spells which kill somebody directly, and there aren’t many ways to destroy horcruxes.
The first which we see is the venom of a basilisk, the only known cure for which is phoenix tears, which are exceptionally rare. Harry uses this to destroy Tom Riddle’s diary. (However, the horcrux inside Harry isn’t killed because he’s saved by the phoenix tears - also, see Q&A here). Hufflepuff’s cup is destroyed by a basilisk fang as well.
Later, we learn that goblin-forged metals (such as the sword of Gryffindor) can absorb properties that strengthen them. Since the sword was used to stab the basilisk, the blade became imbued with basilisk venom, and the sword was then used to destroy Gaunt’s ring, Slytherin’s locket and the snake Nagini.
Another horcrux (Ravenclaw’s diadem) is killed with Fiendfyre in the Room of Requirement. It’s implied that this is quite a complicated piece of magic, and difficult to control (witness Crabbe’s death), but is sufficiently deadly to destroy a horcrux.
The killing curse seems like a good candidate as well: this destroys the horcrux inside Harry, and we’re told that magic can’t reverse its effects (so it meets the criterion). Harry was a special case (a horcrux Voldemort never intended to make), so it’s possible this wouldn’t always have worked. This is never fully established.
That covers the seven horcruxes, so those are the only confirmed ways to destroy a horcrux. Speculating briefly, there are a few other creatures which are fairly dangerous: Dementors, Nundus and Manticores. I would guess that if any of those went for a horcrux, they could destroy it as well, but this has never been addressed in the canon.
The confirmed methods of destroying a Horcrux, according to Book 7, are to use basilisk venom or an object imbued with it, and magical Fiendfyre -- on inanimate Horcruxes. To destroy a Horcrux that is a living thing, one only has to kill the "host."
Hufflepuff's cup and Riddle's diary are destroyed by stabbing with basilisk fangs; Gaunt's ring and Slytherin's locket are destroyed with the venom-imbued Sword of Godric Gryffindor; Ravenclaw's Diadem was destroyed by Vincent Crabbe's out-of-control Feindfyre spell and the two living Horcruxes -- Nagini and Harry himself -- were killed by Neville Longbottom and Lord Voldemort respectively.
But I have to take issue with the idea that Hermione knew multiple methods of destroying Horcruxes (if she did, that was in the movies; she did not know in the book). It seems instead that the methods were discovered by accident or experimentation over the course of Chamber of Secrets, The Half-Blood Prince and Deathly Hallows.
I.e., Dumbledore deduced that the Gryffindor sword was goblin-made and absorbed everything which strengthened it, thus that it was imbued with basilisk venom after Harry used it to slay the basilisk.
Obviously, anything covered in basilisk venom can destroy a horcrux, as can fiendfire. The killing curse destroys the horcrux inside of Harry, when used by Voldemort near the end of The Deathly Hollows, but it has no effect on the snake when cast by Ron. In addition to this, Voldemort attempts to kill Harry at the beginning of The Deathly Hollows using a different wand (not the one that is connected to Harry's) but this also fails. When Voldemort finally "succeeds" at killing Harry Potter, he is using the Elder wand. This leads me to believe that the Elder wand can be used to destroy a horcrux. This theory makes sense when you consider the origin of the Elder wand (It was created by death itself), and the fact that the previous attempts to destroy a horcrux using the Avada Kedavra curse failed.
You can destroy Horcruxes by stabbing them with anything imbued with Basilisk Venom (Gaunt's Ring, Slytherin's Locket, Nagini), throwing them into Fiendfyre (Ravenclaw's Diadem), and in most cases, the Killing Curse (Harry). However, when Ron attempts to kill Nagini with the Killing Curse, it ricochets off of her. This is most likely because Nagini was an intentional Horcrux, while Harry was not. Thus, the Killing Curse MAY only work on certain living Horcruxes, and even then only under certain circumstances.
You must put the Horcrux beyond the power of magical reconstitution, magic can't repair certain damage, Basilisk Venom, Fiendfyre, it's suggested that there are other Substances even more dangerous than Fiendfyre that will work but no details are given, and certainly there are curses that leave indelible damage and in the case of the living death is beyond the power of magic to repair so killing curses or mortal wounds.
In the case, that we never see, of a Horcrux made from an object of a sensitive magical nature, like a wand, pure physical damage, and not much of it would probably be enough since such objects are generally irreparable through the use of magic.