Question in the title. There's so many options: are you a beardy vinyl hipster? Prefer throwing on a random Spotify playlist? Got some wax cylinders from the 1920s? Albums, meticulously curated playlists or "shuffle, play all?"
CD or vinyl would be best, but in the modern world, that's alas not practical.
I have a large iTunes library with over 6,000 offline tracks - but it's on an old PC that doesn't play nicely, so I either play that through my trusty old iPod (particularly in the car, where I do most of my listening) or through Spotify. Advantage of Spotify is being able to discover new things, plus it's linked with last.fm a bit better these days, so it updates my forum signature banner
All sorts. I'll use Spotify to listen to new music to decide whether I want to buy certain albums or not. If I like them I'll buy a physical copy - usually CDs as they take up less space and can be converted to a digital copy more easily, although I do like to buy vinyl. I'll then import them into my itunes library which I sync to my phone. I have a load of different playlists on my itunes library to suit all kinds of purposes, although if I'm feeling particularly daring I'll go for a shuffle of all tracks - wild I know!
I stream in my car/on vacation via a Spotify premium account, but I also have an MP3 player for things not on Spotify or for continuous play compilation albums. At home I left most of my CD's when I moved (only kept the absolute favorites) so I play music off my PC for the most part. My car still has a CD player and I keep maybe 10 albums in there in case my MP3 player dies or I don't wanna use Spotify. Although I have been using Spotify a lot more, more recently because its linked to my Last.FM account // audioscrobbler and I like keeping track. My current MP3 player doesn't sync with Last.FM but my old one (Zune) did.
I pretty much exclusively listen via Spotify now unless I'm driving around home and listening to the radio. It's cheaper than purchasing CD's, which is what I did prior to 2011, and I have access to a vastly larger library than I did with CDs.
I tend to jump from Spotify to YouTube most of the time. If I'm sat with a coffee reading or doing some prolonged type of work then I'll stick on a few albums on Spotify, however I am also a cheap bastard and do like the convenience of YouTube for those moments where you are in the mood for a random song. Some vids on YT are pretty high quality considering the compression (obviously not 320 kbps, but still...).
I listen to new music on Spotify (other streaming services are available) but like to curate an offline collection as well mostly so I can buy albums and support the artists I like. Also got a collection of CDs, mostly Christmas presents because I've never quite worked out how to accept music as a physical gift and I still like having a collection on the side. I basically just take the CD home, rip it and leave it on a shelf.