What's the best set of behind-the-head headphones costing in the neighborhood
of $20? By "best" I mean a combination of good sound, comfort and
durability.

Head phones may not cut it...as sumtimes...there are barriers.......
Sorry.........

After doing some searching, here are the results I found:
http://google1-cnet.com.com/Sony_MDR_G72LP/4505-6468_7-6900035.html?tag=pdtl-list
-

The predecessor to the current model(listed above) got a good review
from several reviewers here:
http://www.consumersearch.com/www/electronics/portable_headphones/

It also depends on what type of music one listens to as the bass in
these street style headphones may not be suited to the music you
listen to(ie. rap).

Do note that for about $10 more you can get better sound quality from
regular headsets(although they will be bulkier)

I hope that helps.
Pushkar

The term "best" depends on one's perspective. What's best for one may
not be best for you.

Given your preference, I'd go for Altec Lansing's AHS302:

Altec Lansing - AHS302
STEREO FOLDABLE BEHIND-THE NECK
Headset with Microphone
http://www.digitalriver.com/dr/v2/ec_MAIN.Entry16?V1=621168&PN=1&SP=10023&xid=48205&V5=31031487&S1=&S2=&S3=&S4=&S5=&V2=&V3=&V4=&DSP=0&CUR=840&PGRP=0&CACHE_ID=0

This was my headset before I upgraded to Altec Lansing's AHS502:

Altec Lansing - AHS502
http://www.digitalriver.com/dr/v2/ec_MAIN.Entry16?V1=621169&PN=1&SP=10023&xid=48205&V5=31031487&S1=&S2=&S3=&S4=&S5=&V2=&V3=&V4=&DSP=0&CUR=840&PGRP=0&CACHE_ID=0

The AHS302 served me well with great sound, durability, and comfort. I
was satisfied with the sound quality it provided. The only reason I
changed to AHS502 is because I wanted louder sound output. With your
criteria, I think you'll be happy with AHS302.