In the grand scheme of good sound, I do tend to follow what Olive has found as most important. That means a system with a smooth response and flat DI, extended bandwidth, and I put extra importance on big dynamic range, meaning high clean output capability.
However, I don't care if a speaker has wide bandwidth because subwoofers are a far better way to reproduce bass anyway. For me, the speaker needs to prioritize high output, a flat response, and a flat DI. I also prefer speakers who can maintain the flat DI over a very wide bandwidth, and I prefer elevated DI's. What that means in practice is a speaker whose response remains consistent with the listening axis response over all angles in the frontal hemisphere, but where the volume level diminishes as you move to the sides. What is called Constant Directivity. While I am ok with a speaker that can extend the elevated DI down to 500hz, I prefer it to extend lower if possible, as it can reduce the main speakers room interaction and thus SBIR problems.
I also want a speaker that can easily produce in excess of 105dB at my listening position. In practice, I really like to have a lot of headroom, so I am not bumping up against the limits. That way, when I do crank up a movie or song, the experience isn't ruined by a speaker that quickly falls apart.
Speakers that have a flat response, flat elevated DI, and an ability to reproduce say 110dB rms cleanly are practically non-existent. You have the Gedlee Abbeys and Summa's (I own the Abbeys), you have some Genelec Models, a few DIY options, Legacy Audio's higher end stuff, JBL M2's. The list is small however.