Hi JimJim -
For less than $400 you have access to a range of very, very good Headphones, from the HE-400s to popular Sennheisers etc.
But here are a few facts -
1) You probably "prefer" a certain sound already, and don't know it.
2) The best way to explore your own preferences is to demo as many headphones as you can
3) The second best way to explore headphones is to demo as many as you can
4) Frequency response curves (like those from headphone.com or Innerfidelity.com), are based on "raw data" that has been "transformed" by what is believed to be the way that people actually perceive sound.
5) Presently, companies like Harmon International are revising that compensation curve to reflect what is objectively a "flat response" as well as what is objectively considered an "ideal" or "preferable" response. The way they are doing this is with trained/untrained listeners and it is a welcome change. An end result appears to have been some new products (the NAD HP50, Focal Spirit Classic/Professional, PSB M4Us I believe) notable for their "neutral" (i.e. "flat") frequency response and overall awesome sound (closer to "good good speakers in a good room"). I haven't gotten my hands on any of those, but my interest is high - I tend to like flatter-sounding speakers.
The point is, frequency response curves in the headphone space will never tell the whole story. Especially at the moment! So, when someone talks about "treble spikes" or "recessed mids", they are typically responding to the information as presented in current frequency response plots, which in any case is based on now out-of-date response curves. Others will claim it is based on their deep subjective knowledge, but as it is subjective, it may not apply to you.
6) You probably shouldn't buy on frequency response curves alone (hence this forum can be helpful to get the users' opinions)
7) The HE-400 sound their best on a good amp, if you don't want to get an amp, buy something else! (Many, many headphones work fine without one).
8) The HE-400 are not "flat" headphones, and were not impressive to many people on a FIRST listen. They get better with time. They get better on velour or modded pads, and with a slight treble cut (3 dB around 8-11 kHz).
9) "recessed mids" refers to the "upper" mids which is really part of the "treble" area if you break down the bass/mid/treble frequencies a bit differently (anything over say 3 khz is really getting into the treble IMO as I think of midrange as capturing most of the human voice). The HE-400s actually have a great midrange IMHO, and my more "mid forward" HD595s do not satisfy me THAT much more in that area (and in fact can be a tad "hot"). In any case, the HD595s got nothing on the HE-400s bass (which is both more extended and more textured).
10) Your favorite music will dictate, to an extent, your satisfaction with any headphone. That booming bass, or "sparkling" treble may be distracting if you love genres like classical or jazz. If you are an "everything" person like me, gravitate toward headphones that are primarily neutral. That will tend to sound good despite the variety of "distorted" or "high resolution" recordings out there. The HE-400s are somewhat "neutral" to my ears as they are amazing for jazz, and good for classical. They falter somewhat in "revealing" the clipping and popping and static of screwed up modern pop recordings (particularly when the vocals are too "hot", e.g. most female pop recordings I've heard).
All that said, if you are new to the hobby, I recommend spending a lot less. I went from $50 to $100 to $250 headphones before spending $400(!) on the HE-400s. There are many excellent products from reputable manufacturers in the $200 to $300 dollar range and I am sure they will produce 90% of the sound you might be looking for. I honestly started getting "buyers remorse" after getting into the $400 range. That is just a lot of dough for what is not necessarily the "end all be all" in sound and I started definitely getting "diminishing returns" (improvement in sound not as great as rise in cost). Also, I was ultimately a bit let down a bit in the comfort/quality of parts with the HE-400. I would recommend them, but only if you really got a hankering for planar magnetic (of which they are the most affordable I know of). That is what happened to me!
This is a great hobby, but unless you absolutely CANNOT blast speakers where you are, I would enter it with caution as GOOD information is tough to find.