I am noticing something both on my iPad 2 and iPhone SE running iOS 9.3.2 that I don't recall seeing until today. When I do a spotlight search, it shows a list of results from different apps, and then at the bottom options to search web, App Store, and maps. But now, if the search is such that there are no results from apps, it won't show those last three search options either.
Also, if relevant, a search that brings up just one top hit of an app and nothing else, also does not show options for the web/app/map search. Also, spotlight might start showing a result, like of a note I'm typing, but as I keep typing just a little more of the title of that note, then all results disappear.
And when offline, this problem goes away. Even about the disappearing notes search result problem.