I'm not an expert on this stuff but will give my opinion...
Using good headphones, regardless of the source of the music, will improve your listening experience. Of course, if you listen to to FLAC using an external sound card or DAC then it will sound better, but I don't think it's completely necessary to enjoy a good set of headphones. Even listening through streaming services on your laptop or phone, a good set of headphones are going to make things sound better.
Having said that, there can be a dependency on what type of headphones you go for. If they have particularly big drivers then a phone might struggle to power them properly which would result in a less than optimal performance but this should only really be a concern if you're looking at super expensive headphones used in studios and stuff.
Managed to get hold of a pair of SoundMAGIC HP151s for £70 off eBay, an absolute steal considering everywhere else I've looked they're around the £120 mark. Cannot wait to get them, I read very good things,
Been using Senheiser HD 598's since 2014 - got really tired of the massively long cable, so I've asked my girlfriend to get me some wireless AKG Y500 headphones for Christmas. They seem to be one of the best head phones around at the 100 quid price point.
PS on the subject of in-ear headphones, I've never been able to enjoy any of them, cleaning, itching, drops in sound, none of them have ever worked for me.
you can get wireless ones and what not. I also bought a usb soundcard for my pc which has made things sound bigger too. I put on THE GREAT BIG NO by Lemonheads yesterday and the bass line sounded so sexy, best thing that ever happened to me.
My soundmagic e10s are still going strong after god knows how many years, but I am treating myself to an upgrade and wanted some decent wireless ones.
I've spent hours researching over the last few days and plumped for these in the end. For quality vs cost it seems they are unbeatable and have been for a couple of years. Previous small gripes seem to have been addressed with recent firmware updates. And the DSEE HX pricessing feature, that upscales compressed audio "making it comparable to hi-res audio files" was too good to resist.
It came down to a choice between the Sonys and the Sennheiser Momentum True Wireless 2 (also considered some Cambridge Audio, B&O and Grado options) but it seems that there's negligible if any improvement over the Sonys for an extra €100.
There's a three month free trial of Tidal with them and after reading about their hi-res files and even better their original master recording quality files I gave it a go. The sound that a couple of little bits of plastic coupled with a 9000 kbps bitrate can produce is baffling and a bit astonishing.
Not cheap, but at €160 not horrificly expensive either. And seemingly worth every penny.