Do you guys use the retire button or do you have a retirement account? Or do you keep the ones that mean the most and their offspring and go from there? What about the other lines that never made it? 

All of mine are on one of two retirement accounts.

I have so many retired animals, I have one for retirees of current lines, and one for breeds I used to do, animals I've rescued from the HA, etc.

I also have so many retired animals, I keep them all on one account. I also like to keep all my animals to preserve their lines and all my hard work. xD

If I have offspring from the animal, I keep it forevverrrr on my retirement account. 
If the animal has no offspring, but is at least 3rd generation, I'll probably move it to my retirement account for memories, plus I may decide to do a retirement breeding and "bring them back".

I hit the "retire" button on any backups that do not sell. (I do not think I have ever retired any animal except for backups that did not sell.)

I think part of the fun for me is that there really is no right way to do it. I'm interested to see everyone else's response to this post to see how they decide to manage their hoard xD