i have quite a few flylines. I've been a bit of a nred on the issue, and I've literally cut up hundreds of lines and shooting heads, modifying, tweaking, sometimes improving, more often ruining them. But it's fun and I've learned a lot. But, hence, I have a lot of lines, many scraps, dozens of shooting heads and a lot of full lines as well.
i always store them clean, and that means cleaning them in luke warm water wit small handful of soap flakes (soap flakes are very mild on the lines, which is not the case with washing up liquid for instance). Soak a cloth and run the fly line through until it leaves no dirt on the cloth. Alternatively, use one of the industry's own, small very mild abrasive pads, they also work really well.
Wipe the line through a cloth with clean, luke warm water, and allow to dry. On floating lines, I always apply some line dressing (you could try Shoot, for instance). Just because the line then feels new and really slick when you roll it back onto the reel.
Coil the line on a line winder and secure with some big pipe cleaners and then put it in a zip lock bag.
I then mark each bag with a Dymo printer (I used to use permanent markers, but they rub off and a box full of lines with the identity rubbed off is exciting to some, a disaster to other, depending on the number of lines you have :-).
i have a large zip lock bags that store all the lines on each line weight I have.
Ive always stored my lines like this, and especially the cleaning part takes some time, but it really keeps the lines singing. I think my oldest lines awe 15 years old,mane they still fish well. Some lines I only use once a year, so it's nice to know that they're well stored and in good shape the next time i need them.
Have a great weekend!