I am rather partial to my HiFiman Sundara's, there are several others who agree. Once again, slightly over budget but well worth the investment.
IMHO you will get much less signal degradation with a wired connection than BT. Bluetooth requires the audio signal to be converted to a transmittable format, transmit the signal, receive the signal, decode than amplify the signal.....There is a lot more going on there than with a simple wired connection.
If you want portability and to HEAR music while doing "x", get a pair of BT earbuds(there are some very nice, inexpensive options) and pair them with your phone. If you want to LISTEN to the music, get a pair of wired cans. Just my nickels worth.