Specifications are important to me, but they need to be of a relevant sort.
On sources:
Specifying something like THD+N is generally not terribly useful. This is for a few reasons:
1. It doesn't tell me distribution. Distribution is generally more important than simply the summed harmonic amplitudes given as a ratio of the total amplitude. Some harmonics are considerably less audible than others, and some are considerably worse sounding than others, so it's important to know distribution.
2. Simple sine tests don't really stress a system in a way that resembles audio reproduction. A lot of things can pass a simple sine test even if they stink in IMD or DIM/TIMD type measurements. It's important to, at a minimum use duotonic (e.g. standard IMD tests) and, really, use multitone tests at a minimum in order to get something useful.
3. Often it's specified at 1KHz, which is a pretty easy test to do well on. 20-20KHz sweeps should be done at a minimum for the standard sine test, though as noted above that isn't sufficient.
FFTs are pretty much where it's at for solving the distribution problem. Multitone takes a more expensive piece of equipment but something like a dedicated audio analyser of good quality can probably do it, or a really nice function generator.
I also, personally, like to know in detail which parts are used. I don't know how useful that is to people generally, but since I've used so many parts in audio applications now I have a pretty decent idea of what I do and don't like in terms of things like op-amps, filter caps, bypass schemes, digital filters, and d/a converters.
On headphones:
I definitely like to know nominal impedance and sensitivity. It's important in terms of knowing what kind of power requirements I'll need...
Aside from that, graphs like what headroom provides can be somewhat useful, but giving a full account of a headphone in terms of specification probably would require something like distance from driver to ear canal (typical distance anyhow), size of the space within the pad, some measure of phase response, maybe some distortion measurements like those noted earlier, and relative amplitude in each direction of the driver (as, in my experience, this does affect what it ends up sounding like).
Headphones are kind of tough to effectively measure just in terms of logistics, and the factors in play require a fairly elaborate rig be put together in order to really be effective. That said, I imagine major companies do use at least applicable technology for the task during the process of designing the headphone, so it's plausible this sort of regimen could be accommodated.
Amps:
Basically the same deal as sources; the same concerns apply. However, it should be tested with different loads as well.