The go-to-way of designing a search experience in today's apps and sites seem to be putting it easily accessible, right up top on your product. Top right corner is probably a quite dominant solution. I think this is great. Even if the product isn’t one where searching is a primary function (even though most products today probably is), it's usually a nice experience to have it extremely accessible.
On the app side, only the search heavy ones (like maps, app store etc.) have a visible search box right up top and center, as opposed to an icon or button. Makes sense since searching is really primary to these kind of apps. Other non-searhy apps usually sticks with the magnifying glass icon somewhere in the title bar. Desktop web sites on the other hand can more easily have the full search box (rather than an icon/button) already available and visible, due to more real estate, even though they are not really search-primary.
To me, the fact that the Facebook app (at least on Android) have a search box on the default home view is a bit unnecessary. Maybe they like to promote their search feature (as they have put some time and money here), otherwise they could easily just put the standard search icon somewhere in the interface.
Facebook search up top (android):
Another design that gets thrown around is the search-in-navbar. Instagram among others use it for instance. In previous versions they had a “discover”- button in the navbar instead of a search icon. I wasn’t particularly fond of this solution as it confused me a bit on the rare occasion when I wanted to search something on Instagram. I don't believe the actual view has changed as you can both search and browse here still. They simply changed the icon from a discover-ish compass to a search-ish magnifying glass. Good choice.
Instagram bottom navbar (android):
The one design I fiercely detest is putting your “search” in a menu or drawer. Spotify - I’m looking at you. Even though two taps doesn’t sound like much of a struggle, my mind always stumble when I want to search some artist or song. And spotify would be an app you could categorize as search heavy. I don’t know why Spotify have decided about this design for their app (again at least on android). Maybe they consider their business focus to be elsewhere? They do put features like “Your music”, “Discover” and “Browse” in heavy spotlight, so it’s possible they want to downplay searching.
The Spotify search in a drawer (android):
One of the reasons I dislike this is the fact that “Search” isn’t really a view. It’s not a piece of content as the other options in a menu usually is. Search is merely a function. Something that effectively is not designed. I find it to actually be quite a simple and generic thing. Putting it in a menu like that signals that it is a view like any other.