Audio Technica ATH-M40x used to be a standard for <$150, unless you’re looking for a specific signature. Also gonna throw out the classic Beyerdynamic DT-770 pro for that price.
It’s been years since I had my pair (gifted them to a friend), but @JAnonymous5150 suggestion of the K553 mkII is fantastic if you can get it for that price. A bit mid-centric, but I really liked mine when I had them.
Do you like a more “attenuated” sound or a “flat” frequency response? My first set was ATH M50x’s in that $$ ballpark, so I recommend those despite a musician oriented, honest FR. Still fun in my opinion, but I prefer less attenuation than boosted headphones usually offer.
AKG K553MkII. You can get the renewed/refurbished ones on Amazon right now for $138. The non-refurb version has gone back up to $229 after going down to $139 for about a month so if you wait a while it’ll probably drop back down. That said, every renewed product I have purchased from Amazon has been in like new condition and they’re still backed by Amazon’s guarantee so they’re a great way to save some cash. I personally think they’re one of the best closed backs available under $400. They’re also significantly better than the K371 that people rave about all the time in both build quality and sound quality. They’re a great neutral closed back with plenty of detail and one of the largest soundstages you can get in a closed back for anything near this price.
I would avoid the M40x and M50x. Not only is their sound shit, with V-shaped sound signature (moderate on 40, severe on 50) and very grainy treble, but the Audio-Technica M series remains the most uncomfortable pair of headphones I’ve ever worn.