Frequency response is the really the most meaningful tech spec. The larger the range, the more sounds the phones can reproduce on the low and high end. However, that does become less meaningful when listening to heavily compressed and processed sources. Also, I would guess that all the options you are considering have the same frequency response, so might not really be a consideration.

Bluetooth version is the next consideration. You want the newest version available, or at least no older than the version your source device runs. The higher versions provide more bandwidth, so theoretically less compression during transmission. It seems common to use older chips in more budget friendly headphones in order to save cost. The average user probably doesn't care, but if you are concerned about sound quality, it can be a factor.

Honestly, your source material is probably more of a hindrance to quality than your equipment. Compression is the enemy. Unless you are going to get into lossless codecs and/or analog sources, you won't be pushing the limits of the headphones you are considering. This is not to say you shouldn't get a nice pair, more just a fair warning that you're not gonna get a major quality boost from just changing your headphones.