How does everyone store their buttercream iced cakes overnight? fridge? Countertop? In the basemen that's cooler? 

In the fridge.

I put them in the fridge even though I use ABC that is technically safe to be left out for 3-5 days at least.  It really depends on the kind of buttercream you are using.  An SMBC technically shouldn't be left out at room temp for more than 4-5 hours, but I'm sure there will be people who will say they leave it out for days without a problem.