Asking for headphone suggestions won't get you nowhere. This matter is so subjective, it's hard to decide from other people's opinions.
That being said, i'm going to try to narrow it down for you a little bit.
I would first want to decide on the form-factor. There are two main forms there, over-the-ear headphones (or cans as they are called) and in-ear-monitor (or IEM) which fit inside your ear. Choosing between the two is just a matter of personal preference, as both can sound the same. I've personally went for IEM as i find them easier to carry around and they look better for me.
Second, you must decide on your budget. If money ain't a thing, then by all means, get some custom IEMs or some expensive cans, but since you're asking here i guess that's not the case. Therefore, i can only make one suggestion:
A decent pair of headphones starts at above 100 EUR. Anything lower will most likely not bring any discernable improvement. Now, since a 100+ investment is not a small amount of money, i recommend, whichever way you take, (cans or IEMs) to get ones that have a replaceable cable. Simply put, your headphone will surely outlast your cables, which will break, tangle, snap etc., and you'll be stuck with a worthless pair of earphones that cannot be repaired.
If you're going for IEMs i would recommend taking a look at Shure SE series. The entry model SE215 sells for around 100 EUR and sounds really good compared to the stock ones. They also come with replaceable cables.
If you want more you can also upgrade to the higher end models, but to be honest, it's not worth the money. I bought the SE215 and SE535 and the difference between them is so small it's not worth 200 EURO's...
Regarding a DAC, i haven't gotten mine yet, but from what i've read, on high sensitivity IEM's such as Shure's the improvement is not so big as on low sensitivity headphones (such as cans). A good cheap DAC would be Fiio E17.
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