After bass comes mid frequencies, or mids. Check vocals and stringed instruments. They should sound like the original. If you listen carefully you'll find that these elements don’t sound at all like the original on bad speakers. Parlance such as ‘punch’ and ‘crispness in the sound’ are often used, which are all subjective terms that mean this portion of frequency is soothing. The checking is useful because music can get fatiguing to the ear after a while if it's spiked up or something.
When it comes to highs, it’s easy. There is a very thin line between ‘bright’ sounding and ‘shimmer’. Bright sounds are the excess treble, and the irritating ‘hiss’ sound in everything. Vocals, drum cymbals, etc. must be checked to see whether they are represented well.
The next factors are spaciousness, timing, detail of the softer instruments etc, which I would call secondary checks. Play a live performance and check the positioning of the players in your virtual soundstage that is created in between the speakers. Musicians who stand at the back should sound that way; it should not be 2D and planar. Drum beats should be precise and clear, and variations in volume between different instruments also should be represented the way it was recorded. Or what’s the point?
Headphones
Headphones deal with sound in pretty much the same way, only the amount of data lost in the wind is low since the contact with the ear is so close. So you approach sound coming through headphones much more personally. What you should look for is a ‘full’ sound. By this I mean you need to check for clarity and preciseness, which is the essence of the sound.
A good pair of headphones can make or break the experience. That’s why after buying an MP3 player, if you need to get more out of the deal, you should invest in a good pair of headphones. All the frequencies are well adjusted, and the sound envelops more environments than you can generate on low output speakers.
The bass should be tight, it may have a lot of punch, but you will notice that good headphones can reproduce lows better and more accurately. At their peak you will a notice a certain jarring. This represents bad sound. With in-ear technology you can get a tighter bass response.
The mids should always be independent of the rest of the sound, and the highs should be well-adjusted. When a song is recorded, the highs are kept on their best behavior. Too much means rickshaw sound.
To check a speakers frequency response.
