What I look for is simple: comfort, durability, isolation (in both directions), neutrality. In that order.
Comfort - I swear by circumaural designs. Some people apparently don't have a problem with pads resting on the pinna, but I find this painful after an hour or two. Isolation is also worse with supra-aural designs.
Durability - most headphones break quickly. The cables are usually the first to go, around the strain relief on either end. I've had some pairs that used a 1/8" minijack on one of the earpieces, making the cable replaceable - until the jack failed and couldn't be replaced without breaking open the earpiece. Good headphones will survive any reasonable drop; really good ones will survive being drop-kicked.
Isolation - if you can't hear the signal over environmental noise, why bother monitoring? If you're recording or working in a theatrical environment, you don't want your headset audio leaking out either. In an office setting, nobody wants to hear your music, no matter how awesoem your jams are. Closed-ear designs are the way to go.
Neutrality - Relatively flat frequency response. But this is way down the list of priorities for me. A great sounding set of headphones that breaks after 2 months, or is uncomfortable to wear, or doesn't block the sound of your workstation's fans is not a great set of headphones.
tl;dr - Beyerdynamic DT250 is my headset of choice. My current pair is almost 15 years old and the only thing that's failed on them is the pleather on the headband, which has fallen off its fabric backing. Still on the original earpads and cable, and they're as good as the day I bought them. Stay away from consumer brands and models; like fast-fashion clothes, they are not built to last.
(This is also why you should buy your cookware from a restaurant supply store, not williams-sonoma.)
I purchased a set of HIFIMan cans with planar drivers on sale for around $500, and I haven't looked back. They sound better than any other >$1000 cans I've listened to, bar none.
That said, they offer no isolation whatsoever, so you do need a quiet environment in which to use them.
The rest are simply concerned with scientific measurement of the system. Does it produce a reasonably flat sound in frequency and phase that remains undistorted at volumes at which they listen to music.
The latter camp will highly rate headphones (or any other piece of audio equipment) that can produce a the full bandwidth of human hearing while keeping distortion low over a useful range of amplitude.
Comfort, cost, and construction quality would also be considered in the ranking.
On the whole, though, there absolutely is a sort of "audiophile" who spends thousands and thousands of perfectly good dollars on completely ridiculous paraphernalia, because a combination of placebo effect and escalating commitment convinces him that it makes his music sound better. I don't have time to dig up a complete selection of background for you, but here's a good place to start, with a fair selection of jumping-off points for further study:
http://goodmath.scientopia.org/2011/12/30/audiophiles-and-th...
The major difference between the fellow described in this article, and the more usual sort of placebo audiophile, is that this fellow happened actually to be an electrical engineer, and thus capable of demonstrating to himself that he was full of it. Unfortunately, such acumen seems to be rather rare.
... oh wait.
The most telling parts are about getting the details from faint instruments, especially at lower volumes. For example the opening part of Beethoven's 9th has very faint sound of violins, and it's often missing on consumer-grade speakers. Ideally I would also compare how wide is the stage and thus how well the instruments are separated, but I don't have the budget for that kind of equipment, if that exists at all.
For my budget, I ended up getting a pair of studio monitors Event 2020bas for speakers, and Etimotic Research as in-ears headphones. YMMV.
I'm not sure any headphones can give you a proper 3D sound stage of the sort that you can get with good loudspeakers (especially the LS3/5a). For that, I use recordings of Shostakovitch's string quartets to see how well I can locate (and distinguish between) four instruments.
Although I'm not fond of it as music, Ravel's Bolero is also a good test track for headphones, because of the way the same theme is repeated by many different instruments.
I use Audio-Technica ATH-ANC7b noise-cancelling headphones, but I don't think there's one headset that suits everyone, any more than there's one pair of speakers, or one car....
Any particular recording you would advise? I mean a specific record of known-good quality, preferably on iTunes, Amazon, or similar. I often find myself spending a lot of time listening to different records of the same piece, just to make sure I get a good one.
One strike against this set is that most or possibly all of the CDs have been produced from LPs. Having bought the vinyl versions (twice!) many decades ago, this is good for me but not so good for you.
http://www.amazon.co.uk/Shostakovich-String-Quartets-1-13-D/...
One thing about newer recordings is that they may well be closer-miked and not have such a "natural" sound stage.
I'm sure this has been discussed on HN before.