Wired will have better fidelity. Below are all wired headphones I recommend.
under $100 Philips shp9500, Sony mdr6, Sony MDR7506, audio technica m40x.
under $200 Philips x1, Sennhiser hd598, audio technica msr7, sony mdr 1r.
under $300 philips X2, Sennhiser hd 650, Sennhiser hd 600, V-moda m100.