Grado - GH3

Shure 1540s for closed backs and Grado GH3s for open. Consistently fantastic.

I have some budget headphones, grado open back and m-audio closed back. I love the grados for almost everything (especially vocals!) and find them non fatiguing and super lightweight so easy to wear for a long time. But, I often want more bass and/or privacy so then I'll pull out the closed back. OP, why not get one of each and still stay in your budget?

I'm a big fan of Sony. The WF 500 earbuds are a very solid budget option, have used mine constantly for well over 3 years and still hold a charge well. And the 1000 xm5 are great over ear noise cancelling headphones that have decent aftermarket support, like cooling ear pads. I love my Grado open back headphones too, but theyre more for dedicated stereo music listening, and don't think they offer anything Bluetooth anyway.

For rock music, there is nothing else like grado, try to find a secondhand one to fit your budget better, but you really need to try it on your own at least once before buying anything else. It's open back, which mean it will leak sound everywhere and much more suitable for home usage. It also doesn't sound that good for most other genre, but man they do the meanest guitar shred I've ever heard