I would have been following you.
I’d try to see if I could turn myself invisible while riding one of those giant magical eagles. Then I’d bundle up with warm clothing and it’s a straight shot to Mordor. The ring goes on only if spotted and only until I evade capture.
If the eagle turns invisible with me, Sauron has to stop a stealth bomber. Otherwise, I’ll be celebrating my victory by fucking Arwen inside of a day.
If the eagle does not turn invisible, then I’ll need a formation of eagles and the forces of evil will need to figure out which one is carrying me.
But if the OP is asking how I’d screw it up, I’d probably be suspicious that it’s too easy. Maybe Sauron anticipated all this and I’m being manipulated into fulfilling his plans? Or maybe Gandalf wants to DM a longer campaign and will destroy anyone savvy enough to solve the quest that way. Obviously, Gandalf must be killed…
I would have taken “cannot be destroyed except in the fires of Mordor” as a challenge, rather than an invariant. The armies of elves and men would be fighting a last rear-guard action as I had “just one last try” at some elaborate mechanism to smash the darned thing.
Ah, yes, the Jack Black solution. 
Obligatory post of the spoof video from the MTV Movie Awards (mildly NSFW):
My failure would somehow be related to orgies.
If by “stealth bomber”, you mean “stealth bomber carrying radar retroreflectors, a high-power transponder, and homing beacons”, then yes. Putting the Ring on will not only not make you invisible to Sauron, it’ll actively attract his attention.
Put Ring in pants pocket. Lose pants.
Failing that, my sense of direction is horrible. Getting to each of the stops on the journey probably takes me three times as long as it takes the Fellowship, so by the time I get to Mount Doom, Sauron is almost finished redecorating Isengard and is deciding between apricot and eggplant for the drapes.
I mentioned something about putting on the ring when spotted by minions and taking it off after evading capture. If I’m not spotted by the minions, I don’t use the ring and don’t tip off Sauron. If I have been spotted by minions, it doesn’t matter if Sauron notices, because he was just going to send minions anyway.
I used to play a live action War of the Ring game set in a large area of state park area. Basically, the idea was to put respective players/forces in place, and see what happens. Net result: strong evil advantage.
The war was lost in all kinds of interesting ways, though the fastest was dropping the Ring in a river near the Shire, in the first ten minutes of the game, and not being able to retreive it (we rebooted the game, though some Evil players left since we ‘won’). There’s just a lot of things that could go wrong. Tom Bombadil not in the right place? Barrow Wights are a big problem. Aragorn delayed from Rivendell? Better hope Gandalf decided to head this way from Isengard and isn’t trying to meet you later. Split up the hobbits into two groups? Run across just one straggler from the Evil Team – say a goblin with some armor, an orc captain, or a Nazgul – who’s also on his way to Mount Doom, and you get iced. Go with an army? You’re easily spotted, and sooner or later you’ll be outnumbered or Sauron shows up and ruins your day. Go alone on the down-low? Gollum’s a problem, and you need to connect with your team at some point – just try to pick out if those are Gondor Black Guards or black-clad Orcs while you’re hiding under a bush 200 yards away. Should you try to circle south, Isengard is in an inconvenient spot; go north, and there’s about 100 orcs, goblins, trolls, wild men, Haradrim and various hangers-on milling around between you and Mount Doom, just drooling at a chance to chase hobbits.
Good still managed to pull through quite a few victories, usually because of Sauron/Saruman troops infighting or lack of coordination of the Evil team. My first year, Evil was really understaffed. Sauron had lucked out with a top number in the lottery for roles after only playing one previous game, which was technically allowable though extremely inadvisable. Gollum also got an ability that let him grab the Ring from an attacked (tag by hand) hobbit with no consideration of combat – like a sneak attack. Only works on hobbits; normally, tagging means you attack and it works out to points of the characters involved, of which Gollum has next-to-none. For a number of convoluted reasons I won’t explain, Frodo and Gandalf had a two-minute head start on the entire Evil army, including Sauron, from Barad Dur heading towards Mount Doom (which is about a five minute stroll, at best). Sauron had gambled on smashing all of the Good army at once and had left absolutely no guards at Mount Doom. The only exception was Gollum, who lost track of the hobbits in the first ten minutes of the game and had not seen them all day, but who decided to – amazingly – bury himself completely in leaves next to the path between Barad Dur and Mount Doom, gambling that the Ringbearer was with the Good army and that he might get a go if he made a run for it.
Indeed, he picked the exact right spot. Here comes Gandalf, with his tall pointy hat, gray robe, eyeball-high wooden staff, and fake beard, and the whole nine yards, and a hobbit with a little green cloak and a waistcoat. Gollum surges out of the leaves yelling, and the players (understandably) are taken aback and fall out of step. Gollum makes his move. He leaps up and triumphantly makes his tag –
on Gandalf.
Meaning, he just attacked Gandalf. Gandalf the White, at that point, even.
Instant dead Gollum.
Gandalf and the hobbit laugh all the way to the Cracks of Doom. Poor Gollum had some ‘splainin’ to do when Sauron found him. 
But me, in Middle Earth? I’d try to wield the One Ring. No question. I wouldn’t mean to hurt anyone, but it’s just the nature of the thing.
Yeah, that’d probably happen to me, too. I’d hear Elrond and Gandalf yapping about its danger, its powerful allure, telling me not to put it on, blah blah blah, but I suspect I’d fall into Boromir’s trap no matter how hard I tried. At the end of the day I’d probably want to use it against Sauron.
Either that, or I’d fall down dead on the slog through Mordor. I don’t think I’d have Frodo’s iron will to go as far as he did, given how incredibly hard the journey was.
This is me. I’d be all “Gandalf,stop with the ‘The ring is inherently eevilll’ thing. I mean, what kind of medieval Manichean unsubtle philosophy do you have going on, here (or is it some kind of Luddite distrust)? Technology is not inherently good or bad, just the uses of it. Let’s figure out how this puppy works, and put that knowledge to work, while crafting political and economic institutions that minimize the amount of centralized power that could put the technology to socially poor ends coughno hereditary monarchycough.”
Meh. Bloody Swiss demigods. I ain’t impressed. 
Me, I would wear all those beautuful dresses and headwear that Arwen wears. But unlike her, I would flee from the Ring Wraiths on my horse, carrying a wounded Frodo…and gotten stuck, jewelled headband and hair and all, in a tree.
If that didn’t happen, I would jump off my horse after crossing the river, and instead of enchanting the river, I would trip on my dress and fall flat on my face. The Wraiths would only have to collect me, Frodo and the ring and let us dryup a bit before they took us to Sauron.
I was thinking along these lines, but including the Ladies of Rivendell, Lorien, etc…
This is more or less how the Three came to be: A great Elven smith learned how to make Rings from Sauron, and then made some good rings of his own using that knowledge. But he didn’t need the One to do so, and you’re probably not going to be able to improve on what he did. Let’s face it, even with hereditary monarchies, Rivendell and Lothlorien were pretty nice places.
First of all, I wouldn’t be so keen as everyone else is to just go along with Gandalf. Granted, Gandalf is a wizard, knows more than me, etc. But then so is Saruman a wizard! And we all know that listening to Saruman would be a serious error. From my knowledge of ancient lore (e.g. reading the Silmarillion) I would know that even the wise and powerful can a) turn to evil, or b) make mistakes.
An example from LOTR: when Aragorn demands his Palantir from Gandalf, Gandalf is all “are you sure you want to use this?” and Aragorn is like “Who are you to tell me what to do with my own stuff?” That would be my attitude.
I would argue that we should use the ring to help us fight Sauron.
But, I would hope that I would be a little more devious than to fall into the Boromir trap. My plan is: choose some pretty powerful character, to use the ring for a while. This character should be so powerful as to be a threat to Sauron, but not so powerful as to be able to become another Sauron in a short period of time. Ideally you would choose someone who is currently good (so that it will take him/her a while to become evil) but also annoying (so that it won’t be too much of a downer when you get rid of him - keep on reading to see how my cunning plan unfolds.)
Our ring-wielding hero (let’s call him Patsy) does pretty good in the war against Sauron, chasing him from Barad-dûr in the same way that Gandalf and Company beat the Necromancer and chased him from Dol Guldur in The Hobbit. At this time, me and my pals pull a fast one, like the Roman senators did to Jules Caesar: “Sorry, Patsy, old buddy but you might be getting too big for your britches”. A quick thrust of Glamdring 'twixt the ribs, pull the precious off Patsy’s finger, hand it over to Frodo or some similar strong-willed character, and then it’s off to Mount Doom for the Ring. The going should be pretty easy now, with Sauron in hiding somewhere else.
Et voilà, easy as pie.
I’d take the whole “The One Ring can only be destroyed in the fires of Mount Doom” thing as a challenge…
So, the Ring has a weakness to being melted down, lets fire up a forge as hot as possible and drop the ring in…
no luck? okay, try some other form of intense heat to melt down the Ring, dragon’s fire?, Balrog-smashing? no luck?
Okay, so the Ring seems to be resistant to these forms of heat-treating, lets conjure up a small tac-nuke and drop it on the Ring…
Hmm, still nothing? well, lets look at it from a different angle, instead of melting down the Ring itself, let’s use Gold’s chemical properties against it, drop the Ring in some elemental Mercury and have it bond with the gold of the Ring to mask it’s “magical” ability, then melt some iron, add in some Carbon, and Chromium, to create some primitive Stainless Steel, pour the molten steel into a large crucible, drop the Ring in the molten steel, then let the steel block form with the ring inside, heat-treat/temper the mercury-infused Ring-in-the-block-of-stainless-steel, then drop the One Block into the Marinaris Trench (for more than ten minutes)…
…where the One Block will most likely be found by the submerged Megatron, who will use the One Block to find the remnants of the AllSpark, and rise from the watery depths to wreak havoc on Earth, overthrow The Fallen, and turn the Earth into the new Borg Homeworld…
…on second thought, just nuke it from orbit, it’s the only way to be sure
Now that’s a mashup.
I would lose it in the bottom of my purse (I swear, I put things in my purse and they disappear into another dimension somewhere). Or I would put it away in a ‘safe place’ and promptly forget where that place was. OR most likely would leave it behind at the last place we stopped; and, being a passive idiot, would be afraid to speak up and tell my companions, hey, we have to go back to the Prancing Pony, I left the ring hidden under a loose floorboard.
- Is it secret ? Is it safe ?
- Humm… yes. Very, very secret. But not so safe, no.