(1) Servers I own that are physically located in my house.
(2) Servers I own that are co-located at a hosting facility.
(3) (physical) Servers that I am renting from a hosting facility.
(4) Virtual servers that I am renting from a hosting facility.
I assume that most law enforcement officials feel that 2-4 shouldn't need a warrant because they are not located in my house, so therefore I don't get protection from unreasonable search/seizure.
Also, at what point do #3 or #4 start to encroach onto Dropbox-esque territory where I own the data, but they are holding it for me under contract?
As I understand it, things are shakier in case (3) and (4), because your provider absent a contract with you that says otherwise could choose to comply with subpoenas (providers commonly claim that they'll respond to LEOs only if they have a "subpoena or a warrant", as if that was a feature; you should assume that subpoenas are mere formalities and that any LEO can obtain one practically on demand). It's possible that your remedy in such a case would be civil, which would be cold comfort if your goal was to have evidence excluded from a trial.
There are a bunch of lawyers on HN now, and I'm sure one of them will jump in here to correct me.