My hospital asked if I wanted to keep mine. I didn't know what I'd do with it, so I declined. This time I'm planning to eat it, or at least take a bite. (Dh doesn't think I'll do it!
I hear it's standard procedure to give them to the pathology lab for routine studies, then they get tossed. Maybe some hospitals won't let you have it at all.
I hear that most hospitals have a rule of "If we remove it from your body, it's our property." (I guess this does not apply to babies, though some hospitals make you wonder, the way they take them away when they want and make rules about how and when they can be fed!) But if you get your arm amputated there is no way you're taking it home with you. Though I'd like to see that rule changed...
The rule also affects women who miscarry. They want the fetal remains for a memorial service; the hospital puts them in the incinerator.