Best Headphones Under $20
Koss KSC75
The Koss KSC75 features a slightly unusual earhook design. This makes them very lightweight and comfortable. This is nice for wearing them while jogging, working out, and engaging in other physical activities but leads to poor noise isolation. These headphones are well renowned for sounding much better than their price would indicate. They produce a well ballanced sound with just a slight emphasis on the high end, likely due, in part, to the fact that they just sit on the outside of the ears and don't cup the ear like full headphones do. Because they sound so good but are so inexpensive, KSC75s are very popular headphones and people have come up will all sorts of different mods for them.
Tascam TH-02
The Tascam TH-02 headphones are made of plastic but feel sturdy. They are pretty comfortable but can get a little annoying after several hours of continual use. The sound they produce is closer to neutral than practically any other headphone near this price point. Bass sounds excellent, especially considering the price. It isn't really emphasized but has good extension, is deep and punch, and is not muddy, like many less expensive headphones. The treble lacks a little bit of detail but overal the sound is very good.

