I actually just keep mine in a cupboard my cats can't get into (and I have more than 60 flavors- not sure I want to admit how many I have, but let's just say that I have more than 60.) But I have a system, and I can grab any bottle I want in at most three grabs, with the median one grab. I just line them up in categories (strawberry, melon, more than one row of creams, lightest to darkest, etc.) When I want a particular bottle I know where it is, and when I'm done with it I put it back where it belongs- the system is a bit more complicated than that, but that's the gist of it. I'd recommend trying something like that out first because 60 flavors isn't really a lot to manage and you're going to have to at least organize them no matter what you do, if you want to be able to grab one at will. My system has the virtue of being very inexpensive- it costs nothing.
Some people swear by nail-polish racks though.